ANGLE ANGULAR MEASUREMENT. Angle is defined a rotation result from initial side to the terminal side...

ANGLEANGLE

ANGULAR MEASUREMENT

• Angle is defined a rotation result from initial side to the terminal side

• An angle has “a positive” sign if its rotation is anticlockwise

• An angle has “a negative” sign if its rotation is clockwise

• We only talk about the magnitude of angle~ not observe its signs

Initial side

Terminal side

o

radius

radius

arc=radius

What is a radian?

Angle = 1 rad

Why do mathematicians use radians instead of degrees?

How many times does the radius divide into the circumference?

There are 2 radians in a circle.

1 radian = = 57.3o

4

THE TRIGONOMETRIC RATIOS

180 rad

360 rad 2

rad 180 1

180 rad 1

Convert each angle in radians to degrees.

Convert each angle in degrees to radians.

1. 2c

2. 5c

3. 3 c

4. c

5. c

1. 65o

2. 200o

3. 120o

4. 180o

5. 330o

114.6o

286.5o

540o

90o

240o

1.13c

3.49c

Length of an arc using radians

Area of a sector using radians

Two important formula using radians

Trigonometric Ratios

Objectives/Assignment

• Find the since, the cosine, and the tangent of an acute triangle.

• Use trionometric ratios to solve real-life problems

Finding Trig Ratios

• A trigonometric ratio is a ratio of the lengths of two sides of a right triangle. The word trigonometry is derived from the ancient Greek language and means measurement of triangles. The three basic trigonometric ratios are sin, cos, and tan

Trigonometric Ratios

• Let ∆ABC be a right triangle. The since, the cosine, and the tangent of the acute angle A are defined as follows.

ac

bside adjacent to angle A

Sideoppositeangle A

hypotenuse

A

B

C

sin A =Side opposite A

hypotenuse=

a

c

cos A =Side adjacent to A

hypotenuse=

b

c

tan A =Side opposite A

Side adjacent to A=

a

b

Exercise:

• In the PQR triangle is right angled at R happens cos P = 8/ 17…find the value of tg P and tg Q ?

Exercises

1. PQR triangle is right angled at R & sin P cosQ = . Find out ⅗the value

tgQtgP

c

AB

DE

2 ABC triangle is right angle at A. BC=p,

AD is perpendicular at BC, DE

perpendicular at AC, angle B = Q, prove

that : DE=psin2 Qcos Q

Evaluating Trigonometric Functions

• Acute angle A is drawn in standard position as shown.

Right-Triangle-Based Definitions of Trigonometric Functions

For any acute angle A in standard position,

adjacent side

opposite sidetan

hypotenuse

adjacent sidecos

hypotenuse

opposite sidesin

x

yA

r

xA

r

yA

opposite side

adjacent sidecot

adjacent side

hypotenusesec

opposite side

hypotenusecsc

y

xA

x

rA

y

rA

Finding Trigonometric Function Values of an Acute Angle in a Right Triangle

Example Find the values of sin A, cos A, and tan A

in the right triangle.

Solution– length of side opposite angle A is 7– length of side adjacent angle A is 24– length of hypotenuse is 25

247

tan,2524

cos,257

sin AAA

Trigonometric Function Values of Special Angles

• Angles that deserve special study are 30º, 45º, and 60º.

Using the figures above, we have the exact values of the special angles summarized in the table on the right.

Let’s try to prove it !

O

B

A

Y

X

45O

OA=OBOA2 + OB2 = OC2

OA2 + OA2 = r2

2OA2 = 1OA2 = ½

OA = = OB

How about 300 , 600 and 900

Cofunction Identities

• In a right triangle ABC, with right angle C, the acute angles A and B are complementary.

• Since angles A and B are complementary, and sin A = cos B, the functions sine and cosine are called cofunctions. Similarly for secant and cosecant, and tangent and cotangent.

Bbc

A

Bba

A

Bca

A

cscsec

cottan

cossin

Cofunction Identities

Note These identities actually apply to all angles (not just

acute angles).

If A is an acute angle measured in degrees, then

If A is an acute angle measured in radians, then

)90sin(cos

)90cos(sin

AA

AA

)90csc(sec

)90sec(csc

AA

AA

)90tan(cot

)90cot(tan

AA

AA

AA

AA

2sincos

2cossin

AA

AA

2cscsec

2seccsc

AA

AA

2tancot

2cottan

Reference Angles

• A reference angle for an angle , written , is the positive acute angle made by the terminal side of angle and the x-axis.

Example Find the reference angle for each angle.

(a) 218º (b)

Solution (a) = 218º – 180º = 38º (b)

65

665

Special Angles as Reference Angles

Example Find the values of the trigonometric functions for 210º.

Solution The reference angle for 210º is 210º – 180º = 30º.

Choose point P on the terminal side so that the distance from the origin to P is 2. A 30º - 60º right triangle is formed.

3210cot3

32210sec2210csc

33

210tan23

210cos21

210sin

Finding Trigonometric Function Values Using Reference Angles

Example Find the exact value of each expression.(a) cos(–240º) (b) tan 675º

Solution(a) –240º is coterminal with 120º.

The reference angle is 180º – 120º = 60º. Since –240º lies in quadrant II, the cos(–240º) is negative.

(a) Similarly, tan 675º = tan 315º = –tan 45º = –1.

21

60cos)240cos(

Finding Angle MeasureExample Find all values of , if is in the

interval [0º, 360º) and Solution Since cosine is negative, must lie in either quadrant II or III. Since So the reference angle = 45º.

The quadrant II angle = 180º – 45º = 135º, and thequadrant III angle = 180º + 45º = 225º.

.cos 22

.45cos,cos 221

22

Ex. 1: Finding Trig Ratios• Compare the sine, the

cosine, and the tangent ratios for A in each triangle beside.

• By the Similarity Theorem, the triangles are similar. Their corresponding sides are in proportion which implies that the trigonometric ratios for A in each triangle are the same.

15

817

A

B

C

7.5

48.5

A

B

C

Ex. 1: Finding Trig RatiosLarge Small

15

817

A

B

C

7.5

48.5

A

B

C

sin A = opposite

hypotenuse

cosA = adjacent

hypotenuse

tanA = opposite

adjacent

8

17≈ 0.4706

15

17≈ 0.8824

8

15≈ 0.5333

4

8.5≈ 0.4706

7.5

8.5≈ 0.8824

4

7.5≈ 0.5333

Trig ratios are often expressed as decimal approximations.

Ex. 2: Finding Trig Ratios

S

sin S = opposite

hypotenuse

cosS = adjacent

hypotenuse

tanS = opposite

adjacent

5

13≈ 0.3846

12

13≈ 0.9231

5

12≈ 0.4167

adjacent

opposite

12

13 hypotenuse5

R

T S

Ex. 2: Finding Trig Ratios—Find the sine, the cosine, and the tangent of the indicated angle.

R

sin S = opposite

hypotenuse

cosS = adjacent

hypotenuse

tanS = opposite

adjacent

12

13≈ 0.9231

5

13≈ 0.3846

12

5≈ 2.4

adjacent

opposite12

13 hypotenuse5

R

T S

Notes:

• If you look back, you will notice that the sine or the cosine of an acute triangles is always less than 1. The reason is that these trigonometric ratios involve the ratio of a leg of a right triangle to the hypotenuse. The length of a leg or a right triangle is always less than the length of its hypotenuse, so the ratio of these lengths is always less than one.

Using Trigonometric Ratios in Real-life

• Suppose you stand and look up at a point in the distance. Maybe you are looking up at the top of a tree as in Example 6. The angle that your line of sight makes with a line drawn horizontally is called angle of elevation.

Ex. 6: Indirect Measurement• You are measuring the height of

a Sitka spruce tree in Alaska. You stand 45 feet from the base of the tree. You measure the angle of elevation from a point on the ground to the top of the top of the tree to be 59°. To estimate the height of the tree, you can write a trigonometric ratio that involves the height h and the known length of 45 feet.

The math

tan 59° =opposite

adjacent

tan 59° =h

45

45 tan 59° = h

45 (1.6643) ≈ h

75.9 ≈ h

Write the ratio

Substitute values

Multiply each side by 45

Use a calculator or table to find tan 59°

Simplify

The tree is about 76 feet tall.

Ex. 7: Estimating Distance

• Escalators. The escalator at the Wilshire/Vermont Metro Rail Station in Los Angeles rises 76 feet at a 30° angle. To find the distance d a person travels on the escalator stairs, you can write a trigonometric ratio that involves the hypotenuse and the known leg of 76 feet.

d76 ft

30°

Now the math d76 ft

30°sin 30° =

opposite

hypotenuse

sin 30° =76

d

d sin 30° = 76

sin 30°

76d =

0.5

76d =

d = 152

Write the ratio for sine of 30°

Substitute values.

Multiply each side by d.

Divide each side by sin 30°

Substitute 0.5 for sin 30°

Simplify

A person travels 152 feet on the escalator stairs.