1. Inverse Trigonometric Functions 4.7
By: Ben O’Shasky, Luke Gause, and Tyler Gilbert

2. Introduction to Inverse Trigonometric Functions
The Inverse Trigonometric Functions are the inverse functions of the Trigonometric Functions. They are viewed as: sin−1, cos−1, tan−1 . You can evaluate them on your calculator or evaluate them without your calculator which we will be showing through our examples later on. You use the Inverse functions when you have the trigonometric ratio and you need to find the angle that has that trigonometric ratio.

3. Unit Circle (0, 1) (1, 0) (-1, 0) (0, -1) 60º 45º Red= 30º angles
Note:
Cosine is negative in the II and III quadrant.
Sine is negative in the III and IV quadrant.
Tangent is negative in the II and IV quadrant.
(0, 1)
60º
45º
Red= 30º angles
Blue= 45º angles
Green= 60º angles
Purple= 90º angles
30º
(1, 0)
(-1, 0)
X= Cosine
Y= Sine
Y/X= Tangent
(0, -1)

4. Unit Triangles 45, 45, 90 Triangle 30, 60, 90 Triangle 45º √2 60º √3 1
30º
45º 2 1
***Use the 45, 45, 90 triangle when you are dealing with an angle that is a 45º angle. Use the 30, 60, 90 triangle when you are dealing with a 30º or a 60º angle.
Match the trigonometric ratio you are given (depending on if you are using sine, cosine, or tangent) to the angle (X) that you are trying to find.

5. Evaluating Angles Using Inverse
Y=sin-1 X X=sin Y
Example: sin-1 (1/√ 3) = X
Step 1: Figure out which triangle to use
The fraction has numbers from the
30, 60, 90 triangle. Since sine is
Opposite side over Hypotenuse, the
angle we are looking for must match
this fraction.
Step 2: The angle that has an opposite of 1 and has the hypotenuse of √3 is the 30º angle. Therefore the X= 30º
The process is the same for cosine and tangent.
Cosine is adjacent/hypotinuse
Tangent is opposite/adjacent
√ 3
60º 1
30º 2

6. Evaluating Inverse Using Calculator
Steps on a Calculator
2nd
Sin, cos, or tan (depending on what inverse you are using)
Type in the trigonometric ratio.
Make sure you have parentheses around ratio when needed.
Example: cos(X) = 1/2
cos-1 (1/2) = X
Follow the calculator steps
Cos-1 (1/2)
Press “ENTER” to get the angle.
In this example…. 60
Unknown Angle
Trigonometric Ratio

7. Calculating a viewing angle
In order to calculate a viewing angle you need to know two side lengths of a triangle
Ex.) 10 6 x
Arcsin(10/6) = x
X =

8. tan-1(1) cos-1(1/2) sin-1(1/√2) sin-1(1/2) tan-1(√3)
Find the Exact Value Without using a calculator, than check your answers using a calculator.
tan-1(1)
cos-1(1/2)
sin-1(1/√2)
sin-1(1/2)
tan-1(√3)

9. Flash Card Answers
π /4 or 45º
π /3 or 60º
π /6 or 30º

10. Multiple Choice Questions
Find the exact value without a calculator for the ones without decimals and a calculator for the ones with decimals…
Arcsin(√3/2)
Arc sin(-1/√2)
Arc cos(1/2)
Arc sin( ½)
Arctan (1/4)
Arc cos( 3/5)
Arcsin(2/3)
Arctan(1/6)
Arccos(5/6)
Arctan(383/500)

11. Multiple Choice A)60 degrees B) 45 degrees C) 30 degrees D) 90 degrees

12. Multiple Choice Continued
A) 50 degrees
b) 75 degrees
c) 30 degrees
d) 105 degrees
A) degrees
B) degrees
C) degrees
D) degrees
A) degrees
B)72.541degrees
C) degrees
D) degrees

13. Multiple Choice Continued
7. A) π
B) .232π
C).934π
D)1.567π
A) 9.462 B)
C)2.537
D)49.321 A)
B)11.254
C)89.352
D)33.557
A)50
B)75.284
C)60.302
D)37.452

14. Answers A C B D

15. References http.//www.youtube.com/watch?v=WTz6PQIFNek
intro/precalculus-trigonometry/28-the-unit-circle-01.htm
Precalculus Graphical, Numerical, Algebraical (book)