Trigonometry - Math Beaconmathbeacon.ca/supportfiles/M10/trigonometry2010.pdf · Trigonometry STRAND Measurement DAILY TOPIC EXAMPLE 4. Develop and apply the primary trigonometric

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Trigonometry

This booklet belongs to:__________________Period____

LESSON # DATE QUESTIONS FROM

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Trigonometry

STRAND

Measurement

DAILY TOPIC EXAMPLE

4.

Develop and apply the

primary trigonometric

ratios

(sine, cosine, tangent)

to solve problems that

involve

right triangles.

4.1 Explain the relationships between similar

right triangles and the definitions of the

primary trigonometric ratios.

4.2 Identify the hypotenuse of a right triangle and the opposite and adjacent sides for a

given acute angle in the triangle.

4.3 Solve right triangles, with or without technology.

4.4 Solve a problem that involves one or more right triangles by applying the primary

trigonometric ratios or the Pythagorean

theorem.

4.5 Solve a problem that involves indirect and

direct measurement, using the trigonometric

ratios, the Pythagorean theorem and

measurement instruments such as a

clinometer or metre stick.

[C] Communication [PS] Problem Solving, [CN] Connections [R] Reasoning, [ME] Mental Mathematics [T] Technology, and

Estimation, [V] Visualizatio



Key Terms

Term Definition Example

Triangle

Similar Triangles

Theta

Acute angle

Obtuse angle

Right Triangle

Oblique Triangle

Legs

Hypotenuse

Trigonometry

Opposite side

Adjacent side

Sine ratio

Cosine ratio

Tangent ratio

Theta (�)



Why Trigonometry?

There is an application of trigonometry that could solve each problem below.

A student approaches a large Sequoia

tree outside the entrance to the school

and wonders how tall the tree is.

A homeowner wants to cut a new board to replace a decaying roof truss. He can measure the

horizontal distance and the angle of inclination but needs to know how long to cut the board.

An engineer is constructing a Ferris wheel for a downtown park. There are 16 passenger carts.

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P a g e 5 |Trigonometry

Similar Triangles:

To understand what trigonometry is, we need to understand the properties of similar triangles.

Two triangles are similar if…

• They have the same angles

• Ratios of corresponding

Determine if each of the following pairs of triangles are similar. Explain why or why or not.

1.

3.

Trigonometry – June18-11

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same angles.

Ratios of corresponding sides are equal.


2.

4.

∠� � ∠�

∠� � ∠�

∠� � ∠

��

��

��

� �

��

�

Δ�� " ∆��

Mathbeacon.com. Use with permission. Do not use after June 2011



��

�

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The Right Triangle

Find the indicated side length

5.

8.



A right triangle is a triangle with one right angle (

The side opposite the right angle is called the

hypotenuse.

The other two sides are called ”legs”.

The sides of the right triangle form a

Pythagorean Triple. That is, they satisfy the

Pythagorean Theorem: $% & '% � (

Some Pythagorean Triples:

)3,4,5., )5,12,13., )7,24,25., )8,15,17.

Find the indicated side length (nearest tenth) in the following right triangles

6.

7.

9.

10.


A right triangle is a triangle with one right angle (90°).

The side opposite the right angle is called the

The other two sides are called ”legs”.

The sides of the right triangle form a

Pythagorean Triple. That is, they satisfy the %.

., )9,40,41., )11,60,61, ., …

in the following right triangles.



One acute angle is indicated on each of the following triangles. If possible, label each triangle with:

opposite, adjacent, and hypotenuse in respect to that angle. Remember, only right triangles can be

labeled this way.

11.

12.

13.

14.

15.

16.



Observe the three “embedded” similar triangles below. Find the missing information on the right.

17. What are the names of the three

triangles?

∆ ABC " ∆______ " ∆______

18. ∠�� ∠______ � ∠______

∠�� ∠______ � ∠______

∠�� ∠______ � ∠______

19. 78

98�

:

;� 0.7500

<=

9=�

�

>?

�

�

20. Based on the work you’ve done, how do you know all three triangles are similar?

21. Earlier, we saw that similar triangles have the following similar ratios:

(using the two smaller triangles embedded above) ��

@��

��

��

��

�@�

1

2

In the questions above we see that we could write different equivalent ratios: 78

98�

<=

9=�

>?

9>�

:

;AB 0.7500 [TANGENT RATIO]

Write two more sets of equivalent ratios that would be true for the similar right triangles

above. 78

�

9<�

>? �

:

L AB 0.6000 [SINE RATIO]

�

�

�;

L AB 0.8000 [COSINE RATIO]

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Trigonometry of Right Triangles

Since similar right triangles have equ

to find unknown angles and/or side lengths.

These ratios have been calculated and stored in our calculator for many angles to help us solve

problems.

We will use the three primary trigonometric ratios:

Tangent ratio

The Tangent Ratio

For an acute angle in a right triangle, the ratio of

We have seen previously that in

constant despite reducing or enlarging the triangle.

These ratios have been calculated and

problems.

22. Challenge Question

Find the tangent ratio for angle C in


What is O$P�for the triangle to the



Trigonometry of Right Triangles

similar right triangles have equivalent ratios for corresponding angles, we can use those ratios

to find unknown angles and/or side lengths.


rimary trigonometric ratios:

Tangent ratio Sine ratio Cosine ratio

For an acute angle in a right triangle, the ratio of QRRQSTUV ∠W

XYZX[V\U ∠W is called the TANGENT RATIO.

We have seen previously that in similar right triangles, the ratios of the legs of a triangle remain

constant despite reducing or enlarging the triangle.

have been calculated and stored in our calculator for many angles to help us solve

Challenge Question

angle C in the triangle to the right.

Challenge Question

for the triangle to the right?


ratios for corresponding angles, we can use those ratios


Cosine ratio

is called the TANGENT RATIO.

right triangles, the ratios of the legs of a triangle remain

angles to help us solve

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The Tangent Ratio

Remember, the tangent ratio is a ratio

]$P� �A^Â_Òa ∠�

$bc$(aPO ∠�

From the diagram we see that Tan

The ratio of one leg to the other is 3:4 or 0.75.

Find the tangent ratio for the indicated angles below.

Answer as a fraction AND as a

24. Find the tangent ratio for

angle C in the triangle

below.

O$P� �d%

de �

:

;

= 0.7500



the tangent ratio is a ratio involving the “legs” of the right triangle.

From the diagram we see that Tan C = :

;

he ratio of one leg to the other is 3:4 or 0.75.


as a decimal to 4 places.

the tangent ratio for 25. Find the tangent ratio

for angle � in the

triangle below.

O$P� � ______

= _________

26. Find

for angle

triangle below.

O$P�

=

8

15

17 9

�


the “legs” of the right triangle.

Find the tangent ratio

for angle � in the

triangle below.

O$P� � ______

_________

12

15

�

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Answer as a fraction AND as a decimal to 4 places.

27. What is O$P�for the

triangle below?

O$P� �f

12�

35

12

≅ 2.9167

Skill Reminder: Solve the following equations.

necessary.

30. d%

h� 4

33. h

L�

di

%





28. What is O$P�for the

triangle below?

O$P� � _____

= ________

29. What is

triangle below?

O$P� � ____

= _____

Skill Reminder: Solve the following equations. Answer to the nearest hundredth if

31. h

;� 6

32. h

;�

34. %

h� 1.3

35. h

%.L�

� 42

70 24


What is O$P�for the

triangle below?

_____

________

Answer to the nearest hundredth if

� 0.55

� 6

70

� 24

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Given that the ratio of QRRQSTUV

XYZX[V\U


Given that the tangent ratio of


Use your calculator to find tan

Use tan 30 ° to find the length of the



for∠� in ⊿�� is 0.5000, find the length of the missing leg.

Challenge Question

Given that the tangent ratio of Angle F is 0.4000, find the length of the missing leg.

Challenge Question

30 ° (round to 4 decimal places).

length of the missing leg.


is 0.5000, find the length of the missing leg.

is 0.4000, find the length of the missing leg.

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Finding Missing Sides Lengths Using the Tangent Ratio

39. Given that the ratio of QRRQSTUV

XYZX[V\U for∠� in ⊿�� is

0.5000, find the length

of the missing leg.

42. Draw a diagram

illustrating the tangent

ratio for ∠k in right

⊿klm.

Use the tangent ratio to find the missing side lengths

45. The tangent ratio for

the triangle below is

0.2500. Calculate x.

0.2500 �f

4

0.2500

1�

f

4

4)0.2500. � 1)f. f � 1



Finding Missing Sides Lengths Using the Tangent Ratio

is


40. Given that the tangent

ratio of Angle F is


of the missing leg.

41. Use your calculator to find

tan 30

places)

Use


illustrating the tangent

43. The tangent ratio is a

ratio of what two sides

in a right triangle?

44. Can you use the

ratio to find the

hypotenuse of a right

triangle?

Use the tangent ratio to find the missing side lengths to the nearest tenth.

46. The tangent ratio for


0.5325. Calculate x.

47. The tangent ra


2.7321.


Use your calculator to find

30 ° (round to 4 decimal

places).

Use tan 30 ° to find the


Can you use the tangent

ratio to find the


triangle?

The tangent ratio for


7321. Calculate x.

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Use your calculator to find the following ratios to 4 decimal places

48. tan 60° � __________

Find the length of the indicated side to the nearest tenth.

50.

53.



Use your calculator to find the following ratios to 4 decimal places, then solve for x.

__________

49. tan 30° �_____________


51.

52.

54. 55.


, then solve for x.

_____________

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Solving Triangles:

To “solve a triangle” means to find the length of all unknown sides and measure of unknown angles.

56. Explain the steps you

Solve the following triangles.

57.

59.




Explain the steps you would take to solve the following triangle.

Solve the following triangles. Answer to tenths.

58.

60.

The dotted line is an altitude.



would take to solve the following triangle.

The dotted line is an altitude.



Solve each of the following word problems. Include a diagram in your solution.

61. From a point 220 m from the Empire State

Building, a tourist measures the angle of

inclination to the top to be 60°. Calculate

the height of the building to the nearest

metre..

62. A radio tower is 396 feet tall. How far

from the base of the tower is a technician

if the angle of inclination to the top of

the tower is 27°? Answer to the nearest

foot.

63. An airplane approaches a control tower. The angle of depression from the pilot to the tower

is 12°. If the plane is flying at an altitude of 1500 m, how far is the plane from being directly

above the tower (to the nearest kilometer)?

64. At 11:00 in the morning, the angle of

elevation to the sun 58°. A tree in the

school yard casts a shadow of 56 m.

How tall is the tree to the nearest

metre?

65. A student crossing to the west building

casts a shadow on the path. She is 165

cm tall and the angle to the sun is 25°.

How long is the shadow on the path to the

nearest centimetre?

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The Sine Ratio

The sine ratio is a ratio involving the hypotenuse and one leg of the right triangle.

From the diagram to the right we see that sin

The ratio of the opposite leg

to the hypotenuse is 3:5 or 0.60.

Find the sine ratio for the indicated angles below.


66. Find the sine ratio for

angle � in the triangle

below.

_`P� � :

L =

= 0.6000

69. What is _`P�for the

triangle below?

_`P� � ______

= _________

�

3

4

5

24

� 10



sine ratio is a ratio involving the hypotenuse and one leg of the right triangle.

From the diagram to the right we see that sin C = :

L

to the hypotenuse is 3:5 or 0.60.

ratio for the indicated angles below.


67. Find the sine ratio for


below.

_`P� � ______

= _________

68. Find


below.

_`P�

= _________

70. What is _`P�for the

triangle below?

_`P� � ______

= _________

71. What is

triangle below?

_`P�

= _________

8

15

17 9

�

56

�

70


sine ratio is a ratio involving the hypotenuse and one leg of the right triangle.

Find the sine ratio for


below.

_`P� � ______

_________

What is _`P�for the

triangle below?

_`P� � ______

_________

12

15

�

70

� 74

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Finding Missing Sides Lengths Using the

72. Draw a diagram illustrating

the sine ratio for ∠k in

right ⊿klm.

Use the sine ratio to find the missing side lengths to the nearest tenth.

75. The sine ratio for the

triangle below is 0.3788.

Find the length of side x.

78. The sine ratio of a right

triangle is 0.8000. If the

hypotenuse is 20 cm long,

what are the lengths of the

other two sides?



Finding Missing Sides Lengths Using the Sine Ratio

Draw a diagram illustrating 73. The sine ratio is a

ratio of what two

sides in a right

triangle?

74. Can you use the

to find the hypotenuse of

a right triangle?

Use the sine ratio to find the missing side lengths to the nearest tenth.

.


triangle below is

0.2000. Find the

length of side y.


triangle below is

the length of side

The sine ratio of a right

triangle is 0.8000. If the

hypotenuse is 20 cm long,

what are the lengths of the

79. The sine ratio of a

right triangle is

0.4500. If the

hypotenuse is 8 m

long, what are the

lengths of the other

two sides?

80. The sine ratio for a right

triangle is 2.1042. Find the

opposite side if the

hypotenuse is 4 mm.

81. The sine ratio for a right


hypotenuse if the opposite

side is 17 cm.


Can you use the sine ratio

to find the hypotenuse of

a right triangle?

The sine ratio for the

triangle below is 0.6018. Find

the length of side t.

sine ratio for a right


opposite side if the

hypotenuse is 4 mm.

The sine ratio for a right


hypotenuse if the opposite

side is 17 cm.

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82. Explain why the previous questions have no solutions.

What do you notice about the value of ratio where this happens?

Interpret that kind of ratio in the sense of a right triangle.

Use your calculator to find the following ratios to 4 decimal places.

83. sin 45° � __________

86. Find the length of side x.


89.



Explain why the previous questions have no solutions.


Interpret that kind of ratio in the sense of a right triangle.

Use your calculator to find the following ratios to 4 decimal places.

84. sin 60° �__________

85. sin 42




90.



42° �__________

Find the length of side x.

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91.

93.

95. A hiker loses track of her direction and

wanders 22 degrees off course. If she

continues to walk for 12 km to the river

destination, how far away from her actual

destination will she be? (nearest tenth of a

kilometer)




92.

94.

loses track of her direction and

wanders 22 degrees off course. If she

continues to walk for 12 km to the river

destination, how far away from her actual

destination will she be? (nearest tenth of a

96. A lighthouse attendant has a range of visi

of 24 km. A ship on the horizon passes by the

lighthouse. The attendant sees the ship for a

total of 120 degrees. For how many kilometers

was the ship within the attendant

sight? (nearest tenth)


A lighthouse attendant has a range of visibility

of 24 km. A ship on the horizon passes by the

lighthouse. The attendant sees the ship for a

total of 120 degrees. For how many kilometers

p within the attendant’s range of



Draw a scale diagram that would represent each of the following.

97. Draw a triangle that has a the

following:

_`Po �%

L, hypotenuse is 10 cm long.

98. Draw a triangle that has a the following:

O$Pp �d%

L, hypotenuse is 26 cm long.

99. Solve the triangle.

100. Solve the triangle.

101. Find the perimeter of the following

rectangle.

102. Find the total area.

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103. While golfing with his father

Mr. J hits a shot short of a pond.

walks 20 m to his left to a point

directly across the pond from the hole.

The angle between the two lines of

sight is 22°. Find the distance from his

ball to the hole to the neare

a metre.

105. Anya lets out 125 feet of kite string at

Clover Point. The wind pulls the kite string

tight at an angle of 55° to the ground.

Approximate the height of the kite to the

nearest foot .

What assumptions did you make?



While golfing with his father-in-law,

Mr. J hits a shot short of a pond. He

walks 20 m to his left to a point

directly across the pond from the hole.

The angle between the two lines of

22°. Find the distance from his

ball to the hole to the nearest tenth of

104. Find the area of the circle that is not

covered by the shaded

the nearest tenth.

Anya lets out 125 feet of kite string at

Clover Point. The wind pulls the kite string

tight at an angle of 55° to the ground.

Approximate the height of the kite to the

assumptions did you make?

106. The radius of a circular

is 15.43 m. During a flood, a worker in the

water at the side of the tunnel measured

an angle to the centre to be 37°. Find the

depth of the water at its deepest point.

(The water surface forms a chord across

the tunnel.)


Find the area of the circle that is not

d by the shaded triangles. Answer to

circular tunnel in Shanghai

is 15.43 m. During a flood, a worker in the

water at the side of the tunnel measured

an angle to the centre to be 37°. Find the

depth of the water at its deepest point.

rface forms a chord across

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The Cosine Ratio

The cosine ratio is a ratio involving the hypotenuse and one leg (adjacent to angle) of the right

triangle.

From the diagram to the right we see that

This means the ratio of the adjacent leg

to the hypotenuse is 4:5 or 0.8

Find the cosine ratio for the indicated angles below.


107. Find the cosine ratio for

angle � in the triangle below.

(A_� � ;

L

= 0.8000

110. What is (A_�for the triangle

below?

(A_� � ______

= _________

�

3

4

5

24

� 10



sine ratio is a ratio involving the hypotenuse and one leg (adjacent to angle) of the right

From the diagram to the right we see that cos C = ;

L

adjacent leg

80.

sine ratio for the indicated angles below.


sine ratio for

in the triangle below.

108. Find the cosine ratio for


below.

(A_� � ______

= _________

109.

for the triangle 111. What is (A_�for the

triangle below?

(A_� � ______

= _________

112.

8

15

17 �

56

�

70


sine ratio is a ratio involving the hypotenuse and one leg (adjacent to angle) of the right

109. Find the cosine ratio

for angle C in the

triangle below.

(A_� � ______

= _________

112. What is (A_�for the

triangle below?

(A_� � ______

= _________

9

12

15

�

70

�

74



Finding Missing Sides Lengths Using the Cosine Ratio

113. Draw a diagram illustrating the

cosine ratio for ∠k in right

⊿klm.

114. The cosine ratio is a ratio

of what two sides in a

right triangle?

115. Can you use the cosine

ratio to find the


triangle?

Use the cosine ratio to find the missing side lengths to the nearest tenth.

116. The cosine ratio for the triangle

below is 0.1500. Find x.

117. The cosine ratio for the

triangle below is 0.3255.

Find x.

118. The cosine ratio for

angle Φ in a right

triangle is 1.7421. Find

the length of the

hypotenuse if the side

adjacent to Φ is 3.5 cm

long.

119. The cosine ratio for the triangle

below is 0.3482. Find x.

120. Draw a right triangle

with a cosine ratio of

0.3277. Label all side

lengths.

121. Explain why you cannot

have a right triangle with

a cosine ratio of 1.6713.



122. Draw a right triangle with an acute angle that has

an adjacent side equal in length to the opposite

side. Find the cosine ratio for that angle.

123. Draw a right triangle with an acute angle

that has a hypotenuse 50% longer than

the adjacent side. Find the cosine ratio for

that angle.

124. Use a protractor to measure the indicated angle. Then determine the length of side x using the cosine

ratio.

125. Use a protractor to measure the indicated angle. Then determine the length of side x using the cosine

ratio.

x



Use your calculator to find the following ratios to 4 decimal places, then find the indicated

side length to 2 decimal places.

126. cos 60° � __________

127. cos 20° �__________

128. cos 42° �__________





132.

133. 134.

135.

136.

137. Find the length of each leg.

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Solve the following problems involving

138. Find the area of the triangle below.

140. Tucker has two choices to get his ball to the

hole at the 17th at Cordova Bay, go around

the lake or go over it. He decides to go

around the lake as shown on the diagram.

How much farther does he have to hit

ball going around the lake instead of going

straight over it?

(Careful, not a right triangle shown.)



problems involving triangles.

the triangle below.

139. Find the area of a rectangle with a

diagonal of 20 m if the angle between the

diagonal and longer

(nearest tenth)

Tucker has two choices to get his ball to the

at Cordova Bay, go around

the lake or go over it. He decides to go

around the lake as shown on the diagram.

How much farther does he have to hit the

ball going around the lake instead of going

(Careful, not a right triangle shown.)

141. Find the area of the circle that is not

covered by the shaded rectangle to the

nearest tenth of a square unit.


Find the area of a rectangle with a

diagonal of 20 m if the angle between the

side is 25 degrees.

Find the area of the circle that is not

covered by the shaded rectangle to the

nearest tenth of a square unit.

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142. Find the area of the circle not

the shaded rectangle.

144. Find the length of AG to the nearest

tenth of a millimetre.



Find the area of the circle not covered by

143. Find the area of the circle not covered

by the shaded rectangle.

Find the length of AG to the nearest

145. Find the area of rectangle WXYZ to the

nearest square unit


Find the area of the circle not covered

by the shaded rectangle.

area of rectangle WXYZ to the

nearest square unit.



Finding Angles Using the Three Ratios

Recall:

The three primary trig. ratios:

Tangent Ratio: tan � �qV\rUs Qt STYV QRRQSTUV W

qV\rUs Qt STYV XYZX[V\U W

Sine Ratio: _`P� �qV\rUs Qt STYV QRRQSTUV W

qV\rUs Qt suRQUV\vSV

Cosine Ratio: (A_� �qV\rUs Qt STYV XYZX[V\U W

qV\rUs Qt suRQUV\vSV

The stored values in your calculator allow you to find angles using the ratios.

The magic of _`Pwd, (A_wd, and O$Pwd.

Challenge

146. Find the measure of angle A in a

right triangle if tanA = 1.000.

Challenge

147. Find the measure angle B in a

right triangle if sinB = 0.5000.

Challenge

148. What ratio would you use to

find the measure of the

indicated angle?

Challenge

149. Find the measure of the

indicated angle.

Unless otherwise stated,

calculate the measure of

angles to the nearest tenth of

a degree.

Eg. 42.8°

The “inverse trigonometric functions”.

These functions convert the stored

ratios in your calculator to the angle.



Use the Inverse functions to find the indicated angle to the nearest tenth.

150. Find the measure angle

A in a right triangle if

tanA=1.000.

Use the tan-1 button.

Type:

tan-1(1.000)=A

A = 45 °

151. Find the measure

angle B in a right

triangle if

sinB=0.5000.

Use the sin-1 button.

Type:

sin-1(0.5000)=B

B = 30 °

152. What ratio would you use to find

the measure of the indicated

angle? Use the

tangent

ratio:

O$Px �d:

d%

Find the measure of the indicated

angle.

Type: tan-1(d:

d%)= x

x = 47.3 °

153. Find angle A, if

sinA=0.2654.

154. Find angle B, if

cosB=0.2654.

155. Find angle Q, if

tanQ=0.34884.

156. Find angle T, if

sinT=0.7854.

157. Find angle D, if

cosD=1.2654.

158. Find angle U, if

tanU=2.6784.

159. In a right triangle, one

acute angle has sine

ratio of 0.5. Find the

sine ratio of the other

acute angle.


acute angle has cosine


sine ratio of the other

acute angle.


acute angle has cosine


tangent ratio of the

other acute angle.

162. Which of the three

trigonometric ratios (sine,

cosine, tangent) can have a

value greater than 1 ?

163. Draw a right triangle and use it to explain your answer

to the previous question.

On your calculator, find the inverse

functions. Usually 2nd

function

required.

These give us the angle from the ratio!



Working with the ratios to find angles.

Have a plan…

1. Choose the correct ratio {sine, cosine, or tangent}.

2. Fill in the known side lengths into your chosen ratio.

3. Use the “inverse trig. function” to convert ratio � angle.

164. What ratio do the

given sides form for

the indicated angle?

Sine Cosine Tangent

165. What ratio do the given

sides form for the

indicated angle?

Sine Cosine Tangent

166. What ratio do the

given sides form for the

indicated angle?

Sine Cosine Tangent

167. Calculate the ratio

above. Round to 4

decimal places.

168. Calculate the ratio above.

Round to 4 decimal places.


above. Round to 4 decimal

places.

170. Calculate the

measure of angle y to the

nearest tenth of a degree.

171. Calculate the measure of

angle z to the nearest

tenth of a degree.

172. Calculate the measure

of angle � to the nearest

tenth of a degree.

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sides form for the

indicated angle?

Sine Cosine Tangent


above. Round to 4

decimal places.

179. Calculate the measure

of angle ∝ to the

nearest tenth of a

degree.





sides form for the indicated

angle?

Sine Cosine Tangent


sides form for the

indicated angle?

Sine Cosine

177. Calculate the ratio above.

Round to 4 decimal places.


above. Round to 4

decimal places.


angle | to the nearest tenth

of a degree.


angle

tenth of


What ratio do the given

sides form for the

indicated angle?

Sine Cosine Tangent

Calculate the ratio

above. Round to 4

decimal places.

Calculate the measure of

angle � to the nearest

tenth of a degree.

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Find the measure of the indicated angle. Round answers to the nearest tenth of a degree.

182.

184.

186. Find the measure of angle x to the

nearest tenth of a degree..




183.

185.

Find the measure of angle x to the

nearest tenth of a degree..

187. Find the measure of angle

nearest degree.



Find the measure of angle � to the



Solve the following triangles. Calculate answers to the nearest tenth.

188.

189.

190.

191.

192.



Find the area of the following triangles to the nearest square unit.

193.

Area = }XSV×sVTrsU

%

Find base:

O$P45 �A^Â_Òa

1.6

∴ '$_a � 1.6 (�

Area = d.e×d.e

%

Area = 1.2(�%

194.

195.

196.

197. A triangle has side lengths of 8 cm, 7cm

and 12 cm. Find the area of the triangle if

the angle between the 8 cm and 12 cm side

is 34°.

198. A triangle has side lengths of 10 km, 23 km

and 32 km. The angle opposite the 10 km

side is 9.2° Find the area of the triangle.



Applications of trigonometry.

199. A kite stuck in a nearby tree. A child

standing 25 m from the base of a tree pulls

the string tight. If the tree is 30 m tall,

approximately how far is the kite from the

child to the nearest metre?

200. A surveyor measures the angle of elevation

to the top of a building to be 23°. If the

surveyor is 1345 feet from the base of the

building, how tall is the building to the

nearest foot?

201. From the top of a 20 m cliff above a road,

the angle of depression to two approaching

cars is 25° and 40° respectively. How far

apart are the cars to the nearest metre?

202. Two hot air balloons float above the ocean

at a height of 1000 feet. From a sailboat,

the angle of elevation to one balloon is 60°

and to the other balloon is 50°. How far

apart are the balloons to the nearest foot?

203. Two boys on opposite sides of the tree

below measure the angle of elevation to the

top of the tree. If the tree is 175 feet

tall, how many feet apart are the boys?

204. Highway sign shows that the road

descends at a rate of 8%. Draw a diagram

that shows what this means.

If a 3 km section of straight road descends at

this grade, what is the drop in elevation?



205. While golfing with his father-in-law, Mr. J

hits a shot short of a pond. The flag

(hole) is directly across the pond from his

ball. He paces 20 m to the right of his ball

and measures the angle back to the hole to

be 76°. How far is the ball from the hole

to the nearest metre?

206. A hiker leaves base camp travelling due

north at 5 km/h. After two hours, she

turns and travels east. Three hours later,

she sprains her ankle. At what bearing

would a rescue team need to travel to

reach the injured hiker? How far away is

she from base camp? (nearest tenth)

207. A student approaches a large Sequoia tree outside the entrance to the school and wonders

how tall the tree is. He paces 150 metres from the base of the tree

and measures the angle of elevation to the top of the tree

to be 35°. Find the height of the tree to the nearest metre.



208. A homeowner wants to cut a new board to replace a decaying roof truss. He can measure the

horizontal distance and the angle of inclination but needs to know how long to cut the board.

The horizontal distance is 14 feet and the angle of inclination is 24°. Find the distance to the

nearest tenth of a foot.

209. An engineer is constructing a Ferris wheel for a downtown park. There are 16 passenger carts

and the radius of the wheel is 10 metres. How far apart are the passenger carts to the

nearest hundredth of a metre?

210. Find the area of the circle to the nearest

square centimetre. [� � �B%]

211. Find the perimeter of the octagon

inscribed in a circle of radius 8 cm.

(Nearest cm)



212. Find the length of the 25° line of

latitude. The Earth’s radius is 6380 km. Answer to the nearest km.

213. Find the length of the 45° line of

latitude. The Earth’s radius is 6380 km. Answer to the nearest km.

214. Mr. Teespré ’s backyard slopes away from his house towards the beach. The instructions for

his new lawnmower state that the mower should not be used if the slope is greater than 15°.

Being a trigonometry specialist, he extends a level string 125 feet from the base of his house.

From that point, he measures that the distance along the ground back to his house is 130 m.

Is his yard too steep for this

mower?

25˚ N

0˚

45˚ N

0˚



215. A regular pentagon is inscribed in a circle

of radius 10 cm. Calculate the perimeter

of the pentagon. Answer to the nearest

cm.

216. A regular decagon (10 sides) is inscribed

inside a circle of radius 8 cm. Find the

perimeter of the decagon. Answer to

the nearest cm.

217. Find the area of the octagon inscribed in a

circle of radius 8 cm. Answer to the

nearest square cm.

218. A regular hexagon is inscribed in a circle

with a radius 18 cm. What would be the

side length of the hexagon? Answer to

the nearest cm.

219. From a point 15 m from the base of a tree, a woman found the angle of inclination to the top

of the tree to be 45°. Her sister found the angle to be 18° from a point farther away

from the base of the tree. How far away are the two women away from each other? (nearest tenth of a metre)



More word problems using right triangles:

• Draw a diagram.

• Fill in known values.

• Let a variable represent unknown(s)

• Choose an appropriate strategy to solve for the unknown(s).

• Interpret the problem.

220. Solve the triangle given the following. ∆��

f � 9 (�

∠� � 90°

∠� � 36°

221. A firefighter is walking along the river at point C when she spots two fires on the opposite

river bank. She measures the angles below and paces a distance of 300 m from point C to

point D. Point D is directly across the river from one of the fires. How far apart are the

fires to the nearest metre?



222. Anya stands on top of a building in downtown Victoria. From her position, the angle of

elevation to the top of an adjacent building is 47°. The angle of depression to the

base of the building is 62°. She is told that the buildings are 45 m apart. Based on

this information, what is the height of the taller building to the nearest metre?

223. Find the length of diagonal BG in the rectangular prism. Answer to the nearest tenth of a

millimeter.

224. The line of sight from an inflatable boat to the top of an oil derrick is 24 degrees. If the

derrick is 45 m tall, how far is the boat from its base? (nearest tenth)



225. A pilot on a level path knows she should

descend at an angle of 3 degrees to

maintain comfort and safety. If she is

flying at an altitude of 12 000 feet, how

many miles from the runway should she

begin her descent?

226. An aircraft ascends after takeoff at an

angle of 22 degrees. What will be the

altitude of the aircraft after it flies at

that angle for 1200 m? (nearest metre)

227. A hamster scurries up a ramp at a speed

of 1.5 m/s. The ramp is inclined at an

angle of 18 degrees. How many metres

above the ground will the hamster be after

30 seconds?

228. Anya travels down a zip line at 25 km/h.

The angle of descent of the zip line is 11

degrees. How many vertical metres has

she fallen after 3 minutes?

What assumptions did you make?

229. The Earth’s radius is 6380 km. Find the length of the 35° latitude to the nearest

10 km.

230. Find the angle of inclination at the back

of the roof. The “rise” of the roof is 0.9

m. (nearest tenth)



231. A ladder should make an angle of 72° with the ground for maximum safety. If the ladder is 4

m long, how far should it reach up the wall? (nearest tenth)

232. The angle of elevation to the top of a tree, measured on a 1.5 m transit from a distance of 30

m, is 15°. Find the height of the tree. (nearest tenth)

233. Find the value of ‘x’.

234. Mr. J has developed the ideal ice cream

cone. The cone has a slant height of 13 cm

and a diameter of 7.8 cm. Find the angle

that the curved surface makes with the

diameter.

235. Mr. J continues to work on his isolated

surf hut. Below is two-thirds of a roof

truss he wants to complete. Find the

length of wood he must cut (nearest tenth)

to complete the truss. The long side is 8.2 m and the short side is 6.8 m. The angle

between them is 35°.

x 22°

18 cm

18 cm

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236. Both triangles (large and smaller inset)

are isosceles. Find the area of the shaded

trapezoid to the nearest tenth of a square

unit.

238. Find the measure of angle x to




Both triangles (large and smaller inset)

are isosceles. Find the area of the shaded

trapezoid to the nearest tenth of a square

237. From a fire station in central BC, Georgia

travels on a bearing of 37° at 6 km/h.

Shelby leaves the station at the sa

travelling due east at 5 km/h. How far

apart are they after 4.5 hours?

tenth)

Find the measure of angle x to the


239. At 9:00 am, a ship leaves port traveling at

30 km/h on a bearing of 63°.

time, another ship leaves port on a bearing

of 315° at a speed of 19 km/h.

boats stop after two hours, how far east is

the boat at point C?


From a fire station in central BC, Georgia

travels on a bearing of 37° at 6 km/h.

Shelby leaves the station at the same time

travelling due east at 5 km/h. How far

apart are they after 4.5 hours? (Nearest

At 9:00 am, a ship leaves port traveling at

30 km/h on a bearing of 63°. At the same

time, another ship leaves port on a bearing

of 315° at a speed of 19 km/h. When the

boats stop after two hours, how far east is

the boat at point C?



Draw an accurate diagram to answer each of the following questions.

240. In ⊿lm�, ∠l�m � 90°, lm � 12 (� and

l� � 10 (�. Find the measure of ∠lm�

241. In ⊿]��, ∠]�� 90°, ]� � 115 � and

]� � 99 �. Find the measure of ∠�]�

242. In ⊿@��, ∠@�� 90°, @� � 12 (� and

∠@�� 30° . Find the length of ��.

243. In ⊿��, ∠�� 90°, �� 5 (� and

∠�� 12° . Find the length of ��.

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Trigonometry - Math Beaconmathbeacon.ca/supportfiles/M10/trigonometry2010.pdf · Trigonometry STRAND Measurement DAILY TOPIC EXAMPLE 4. Develop and apply the primary trigonometric