9.1 Use Trigonometry with Right Triangles

• Find trigonometric ratios using right triangles.

• Use trigonometric ratios to find angle measures in right triangles.

• Use the Pythagorean Theorem to find missing lengths in right triangles.

• Use special right triangles to find lengths in right triangles.

Pythagorean TheoremIn a right triangle, the square of the length

of the hypotenuse is equal to the sum of the squares of the lengths of the legs.

a2 + b2 = c2

b

ac

In right triangle ABC, a and b are the lengths of the legs and c is the length of the hypotenuse. Find the missing length. Give exact values.

1. a = 6, b = 8

2. c = 10, b = 7

ANSWER c = 10

ANSWER a = 51

Definition

Trigonometry—the study of the special relationships between the angle measures and the side lengths of right triangles.

A trigonometric ratio is the ratio the lengths of two sides of a right triangle.

Right triangle parts.

AB

leg

leghypotenuse

A

B

C

Name the hypotenuse.Name the legs. ACBC ,

The Greek letter theta

Opposite or Adjacent?

• The opposite of east is ______• The door is opposite the windows.

• In a ROY G BIV, red is adjacent to _______• The door is adjacent to the cabinet.

west

orange

Triangle Parts

A

B

C

a

b

c

Opposite sideAdjacent side

hypotenuse Opposite sideAdjacent side

Looking from angle A:Which side is the hypotenuse?Which side is the opposite?Which side is the adjacent?

cab

Looking from angle B:Which side is the opposite?Which side is the adjacent?

ba

Trigonometric ratios

hypotenuseoflengthAoppositelegoflengthsine Aof

ca

ABBC

A

B

C

a

b

c

hypotenuseoflength

Aadjacentlegoflengthcosine Aofcb

ABAC

AtoadjacentlegoflengthAoppositelegoflengthtangent Aof

ba

ACBC

c otangent of ∠𝐴=length of leg adjacent∠𝐴lengthof leg opposite∠𝐴=

𝐴𝐵𝐵𝐶=

𝑐𝑎

secant of∠ 𝐴=hypotenuse

length of leg adjacent∠ 𝐴=𝐴𝐵𝐴𝐶=

𝑐𝑏

=

Trigonometric ratios and definition abbreviations

caA

hypotenuseoppositesin

cbA

hypotenuseadjacentcos

A

B

C

a

b

c

baA

adjacentoppositetan

SOHCAHTOA

Trigonometric ratios and definition abbreviations

A

B

C

a

b

c

𝑐𝑠𝑐∠ 𝐴=h𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒

s

𝑐𝑜𝑡∠ 𝐴=𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒

𝑐𝑠𝑐=1𝑠𝑖𝑛

𝑠𝑒𝑐= 1𝑐𝑜𝑠

𝑐𝑜𝑡= 1𝑡𝑎𝑛

𝑠𝑖𝑛−1

𝑐𝑜𝑠− 1

𝑡𝑎𝑛− 1

Evaluate the six trigonometric functions of the angle θ. 1.SOLUTION

5.= 25= √From the Pythagorean theorem, the length of thehypotenuse is 32 + 42√

sin θ =opphyp =

35 csc θ =

hypopp =

53

tan θ =oppadj =

34 cot θ =

adjopp =

43

cos θ =adjhyp =

45 sec θ =

hypadj =

54

Evaluate the six trigonometric functions of the angle θ.

SOLUTION

sin θ =opphyp csc θ =

hypopp

3.

From the Pythagorean theorem, the length of theadjacent is

=5

5 2√ 5=5 2√

√ (5 √2 )2−52=√25=5.

tan θ =oppadj =

55 cot θ =

adjopp =

55

cos θ =adjhyp sec θ =

hypadj

5= 5 2√ 5=

5 2√

= 1 = 1

45°- 45°- 90° Right TriangleIn a 45°- 45°- 90° triangle, the hypotenuse is √2

times as long as either leg. The ratios of the side lengths can be written l-l-l√2.

l

l

2l

30°- 60° - 90° Right TriangleIn a 30°- 60° - 90° triangle, the hypotenuse is twice

as long as the shorter leg (the leg opposite the 30° angle, and the longer leg (opposite the 60° angle) is √3 times as long as the shorter leg. The ratios of the side lengths can be written l - l√3 – 2l.

60°

30°

l

2l

3l

4. In a right triangle, θ is an acute angle and cos θ = . What is sin θ?7

10

SOLUTION

STEP 1

51.= √

Draw: a right triangle with acute angle θ such that the leg opposite θ has length 7 and the hypotenuse has length 10. By the Pythagorean theorem, the length x of the other leg is 102 – 72√x =

STEP 2 Find: the value of sin θ.

sin θ =opphyp = 51√

10ANSWER

sin θ = 51√10

SOLUTIONWrite an equation using a trigonometric function that involves the ratio of x and 8. Solve the equation for x.

Find the value of x for the right triangle shown.

cos 30º =adjhyp Write trigonometric equation.

32

√ = x8 Substitute.

34 √ = x Multiply each side by 8.

The length of the side is x = 34 √ 6.93.

ANSWER

SOLUTION

Write trigonometric equation.

Substitute.

Solve ABC.

A and B are complementary angles,so B = 90º – 28º

tan 28° =oppadj cos 28º =

adjhyp

tan 28º =a

15 cos 28º =15c

= 68º.

Solve for the variable.Use a calculator.

15(tan 28º) = a

7.98 a 17.0 c

So, B = 62º, a 7.98, and c 17.0.ANSWER

c

𝑐=15

𝑐𝑜𝑠28 °

Make sure your calculator is set to degrees.

Solve ABC using the diagram at the right and the given measurements.5. B = 45°, c = 5SOLUTION

Substitute.

A and B are complementary angles,

so A = 90º – 45º

cos 45° =adjhyp sin 45º =

opphyp

cos 45º =a

5 sin 45º = 5b

Write trigonometric equation.

= 45º.

Solve for the variable.Use a calculator.

5(cos 45º) = a 5(sin 45º) = b

3.54 a 3.54 b

So, A = 45º, b 3.54, and a 3.54.ANSWER

SOLUTION

Substitute.

A and B are complementary angles,so B = 90º – 32º

tan 32° =oppadj sec 32º =

hypadj

tan 32º =a

10 sec 32º = 10c

6. A = 32°, b = 10

Write trigonometric equation.

= 58º.

Solve for the variable.Use a calculator.

10(tan 32º) = a

6.25 a 11.8 c

So, B = 58º, a 6.25, and c 11.8.ANSWER

SOLUTION

Substitute.

A and B are complementary angles,so B = 90º – 71º

cos 71° =adjhyp sin 71º =

opphyp

cos 71º =b

20 sin 71º =a

20

7. A = 71°, c = 20

Write trigonometric equation.

= 19º.

Solve for the variable.Use a calculator.

20(cos 71º) = b

6.51 b 18.9 a

So, B = 19º, b 6.51, and a 18.9.ANSWER

20(sin 71º) = a

SOLUTION

Substitute.

A and B are complementary angles,so A = 90º – 60º

sec 60° =hypadj tan 60º =

oppadj

sec 60º = 7c tan 60º =

b7

Write trigonometric equation.

8. B = 60°, a = 7

= 30º.

Solve for the variable.Use a calculator.

7(tan 60º) = b 7 1(cos 60º

)= c

14 = c 12.1 b

So, A = 30º, c = 14, and b 12.1.ANSWER

A parasailer is attached to a boat with a rope 300 feet long. The angle of elevation from the boat to the parasailer is 48º. Estimate the parasailer’s height above the boat.

Parasailing

SOLUTION

sin 48º =h

300 Write trigonometric equation.300(sin 48º) = h Multiply each side by 300.

STEP 1 Draw: a diagram that represents the situation.

STEP 2 Write: and solve an equation to find the height h.

223 ≈ x Use a calculator.

The height of the parasailer above the boat is about 223 feet.ANSWER

• Find trigonometric ratios using right triangles.

• Use trigonometric ratios to find angle measures in right triangles.

• Use the Pythagorean Theorem to find missing lengths in right triangles.

• Use special right triangles to find lengths in right triangles.

𝑎2+𝑏2=𝑐2

Since, cosine, tangent, cotangent, secant, cosecant

In a 45°-45°-90° right triangle, use

In a 30°-60°-90° right triangle, use

9.1 Assignment

Page 560, 4-14 even, 17, 19, 22-26 even,

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