Inverse Trigonometric Functions
4.7
By: Ben O’Shasky, Luke Gause, and Tyler Gilbert
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.
Unit Circle
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
30º
Red= 30º angles
Blue= 45º angles
Green= 60º angles
Purple= 90º angles
(1, 0)
(-1, 0)
X= Cosine
Y= Sine
Y/X= Tangent
(0, -1)
http://www.youtube.com/watch?v=WTz
6PQIFNek
Unit Triangles
45, 45, 90 Triangle
45º
30, 60, 90 Triangle
√2
1
1
60º
√3
30º
45º
1
2
***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.
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
√3
60º
30, 60, 90 triangle. Since sine is
1
30º
Opposite side over Hypotenuse, the
angle we are looking for must match
2
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
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.
1.
2.
3.
4.
Example: cos(X) = 1/2

Unknown
Angle
cos-1 (1/2) = X

Follow the calculator steps
Trigonometric
Ratio

2nd

Cos-1 (1/2)

Press “ENTER” to get the angle.
o In this example…. 60

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 = 36.870
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)
Flash Card Answers
π /4 or
45º
π /3 or
60º
π /4 or
45º
π /6 or
30º
π /3 or
60º
Multiple Choice Questions
Find the exact value without a calculator for the ones without
decimals and a calculator for the ones with decimals…
1. Arcsin(√3/2)
2. Arc sin(-1/√2)
3. Arc cos(1/2)
4. Arc sin( ½)
5. Arctan (1/4)
6. Arc cos( 3/5)
7. Arcsin(2/3)
8. Arctan(1/6)
9. Arccos(5/6)
10. Arctan(383/500)
Multiple Choice
A)60 degrees
B) 45 degrees
C) 30 degrees
D) 90 degrees
2. A)-π/4
B)-2π/3
C) π/4
D) π/2
3. A) π/3
B) π/4
C) π/6
D) π/2
1.
Multiple Choice Continued
4.
5.
6.
A) 50 degrees
b) 75 degrees
c) 30 degrees
d) 105 degrees
A) 19.048 degrees
B)14.036 degrees
C) 29.476 degrees
D) 68.934 degrees
A) 53.130 degrees
B)72.541degrees
C)92.354 degrees
D)13.458 degrees
Multiple Choice Continued
7. A) 10.341π
B) .232π
C).934π
D)1.567π
8. A) 9.462
B) 23.458
C)2.537
D)49.321
9. A) 51.324
B)11.254
C)89.352
D)33.557
10 A)50
B)75.284
C)60.302
D)37.452
Answers
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
A
A
A
C
B
A
B
A
D
D
References
 http.//www.youtube.com/watch?v=WTz6PQIFNek
 http://www.coolmath.com/precalculus-review-calculus-
intro/precalculus-trigonometry/28-the-unit-circle-01.htm
 http://seyyalsarac.com/2/trig-comics
 Precalculus Graphical, Numerical, Algebraical (book)