# Presentation ```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
•Sine is negative in the III and IV
•Tangent is negative in the II and IV
(0, 1)
60&ordm;
45&ordm;
30&ordm;
Red= 30&ordm; angles
Blue= 45&ordm; angles
Green= 60&ordm; angles
Purple= 90&ordm; angles
(1, 0)
(-1, 0)
X= Cosine
Y= Sine
Y/X= Tangent
(0, -1)
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Unit Triangles
45, 45, 90 Triangle
45&ordm;
30, 60, 90 Triangle
√2
1
1
60&ordm;
√3
30&ordm;
45&ordm;
1
2
***Use the 45, 45, 90 triangle when you are dealing with an angle that is a 45&ordm; angle. Use the
30, 60, 90 triangle when you are dealing with a 30&ordm; or a 60&ordm; 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
60&ordm;
30, 60, 90 triangle. Since sine is
1
30&ordm;
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 30&ordm; angle. Therefore the X= 30&ordm;
 The process is the same for cosine and tangent.
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

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
a calculator.
tan-1(1)
cos-1(1/2)
sin-1(1/√2)
sin-1(1/2)
tan-1(√3)
π /4 or
45&ordm;
π /3 or
60&ordm;
π /4 or
45&ordm;
π /6 or
30&ordm;
π /3 or
60&ordm;
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( &frac12;)
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
1.
2.
3.
4.
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6.
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10.
A
A
A
C
B
A
B
A
D
D
References