5.7

How Can I Find the Angle?

Pg. 24Inverse Trigonometry

5.7 – How Can I Find the Angle?Inverse Trigonometry

You now know how to find the missing side lengths in a right triangle given an acute angle and the length of any side. But what if you want to find the measure of an angle? If you are given the lengths of two sides of a right triangle, can you work backwards to find the measurements of the unknown angles? Today you will work on "undoing" the different trigonometric ratios to find the angles that correspond to those ratios.

5.34 – WHAT'S YOUR ANGLE?Do you recognize the types of triangles shown below? Do any of them look familiar? How can you use information about the side lengths to help you figure out the reference angle (θ)?

45 11 60

5.35 – SAFETY FIRSTMr. Gow needs to build a wheelchair access ramp for the school's auditorium. The ramp must rise a total of 3 feet to get from the ground to the entrance of the building. Mr. Gow has plans from the planning department that call for the ramp to start 25 feet away from the building.

a. Draw an appropriate diagram of this situation. Include all the measurements you can.

3ft

25ft

b. In order to follow the state building code, the angle formed by the ramp and the group cannot exceed 4.76°. Will this ramp meet the state building code? To find an angle from a trig ratio you need to "undo" the trig ratio, just like you can undo addition with subtraction, multiplication with division, or squaring by finding the square root. These examples are all pairs of inverse operations.

When you use a calculator to do this you use inverse trig functions which look like "sin-1", "cos-1", and "tan-1." These are pronounced, "inverse sine," "inverse cosine," and "inverse tangent." On many calculators, you must press the "inv" or "2nd" key first, then the "sin", "cos", or "tan" key.

yes

c. Return to your diagram from part (a). According to the plan, what angle will the ramp make with the ground? Will the ramp be to code?

3ft

25ft

O

A

3

tan25

6.84 no

1 3tan

25

Steps to solving with inverse trig:

1.

2.

Label all sides with reference angle

What two sides do you know?

5.36– USING INVERSE TRIGONOMETRYWrite Use a calculator to approximate the measure of to the nearest tenth of a degree.

A

A

O A=tan−1

30

20

⎛⎝⎜

⎞⎠⎟

A=56.31°

H

O

A=sin−1

14

26

⎛⎝⎜

⎞⎠⎟

A=32.58°

HA

A=cos−1

10

16

⎛⎝⎜

⎞⎠⎟

A=51.32°

5.37 – EXTRA PRACTICEWrite and solve an equation to find the missing side length or angle.

OA

x=tan−1

12

15

⎛⎝⎜

⎞⎠⎟

x=38.66°

H

A cos32° =

x

24

x 20.35

OH

x=sin−1

24

29

⎛⎝⎜

⎞⎠⎟

x=55.85°

O

H

sin57° =

7

x

8.35x

5.38 – FINDING ALL MISSING PARTSSolve the right triangle. Approximate angles to two decimal places.

P PQ QR

180 90 37

53°

OH

A

sin 37 22

x

13.24x

cos 37 22

y

17.57y

P N PN

P =54.78°

OA

180 90 54.78

35.22 122 +172 =x2

433 =x

144 + 289 =x2

433 =x2

17

12

⎛⎝⎜

⎞⎠⎟ P =tan−1

5.39 – CAN I EXIST?Peter cannot figure out what he did wrong. He wrote the equation below to find the missing angle of a triangle. However, his calculator gives him an error message each time he tries to calculate the angle.

Jeri, his teammate, looked at his work and exclaimed, "Of course your calculator gives you an error! Your cosine ratio is impossible!" What is Jeri talking about? And how could she tell without seeing the triangle or checking on her calculator?

adjacent .

hypotenuse

Hypotenuse is always the longest side

Right Triangles Project

Pythagorean Theorem: Given 2 sides

45º– 45º– 90º

30º– 60º– 90º

Sine – S

sin-1, cos-1, tan-1

Your NameBlock#

Cosine – C

Tangent – T

2x x x 3 2x x x

OH

AHOA

Clinometer MeasuresArea of Regular Polygons

sin-1, cos-1, tan-1