1. 8.2: Inverse Trig Functions
© 2008 Roy L. Gover(www.mrgover.com)
Learning Goals:
•Review Special Angles
•Evaluate inverse trig
functions.

2. Review
The special angles are:
6
π
3
π
60° or
4
π
45° or
30° or

3. Consider the first two
special angles in degrees...
30°
60°
Long Side ShortSide
Hypotenuse
Review

4. And in radians:
Long Side ShortSide
Hypotenuse
Review
6π
3π

5. Important Idea
In a 30° ,60° ,
triangle:
•the short side is one-half
the hypotenuse.
•the long side is times
the short side.
3
( 6)π ( 3)π
( 2)π90°

6. Important Idea
In a 45° ,45° ,90°
triangle:
•The legs of the triangle
are equal.
•the hypotenuse is
times the length of the leg.
2
( 4)π ( 4)π
( 2)π

7. Try This
Find the
length of
the
missing
sides: 30°
60°
1
2
1 3 or 3

8. Try This
Find
the
length
of the
missing
sides 1
1
2
45°

9. Important Idea
Many trig functions can be
solved without graphing by
using special angles and
inverse trig functions. A
special angle solution will be
an exact solution whereas a
graphing solution is only
approximate.

10. Definition
Trig
Function
Inverse Trig Function
siny x= 1
sinx y−
=
cosy x= 1
cosx y−
=
tany x= 1
tanx y−
=

11. Important Idea
arcsin y
arccos y
arctan y
In some books:
1
sin y−
1
cos y−
1
tan y−
instead of
instead of
instead of

12. Example
Find the exact value
without using a calculator;
1 1
sin
2
− 1 2
cos
2
− ( )1
sin 1−
−
1
tan (1)− 1 3
cos
2
−  
− ÷
 

13. Example
Find all values of x in the
interval for
which:
2
cos
2
x =
0 360x° ≤ ≤ °

14. Example
Find all values of x in the
interval for
which:
2
cos
2
x =
0 2x π≤ ≤

15. Try This
Find all values of x in the
interval for
which: 1
sin
2
x = −
0 360x° ≤ ≤ °
210 & 330x = ° °

16. Try This
Find all values of x in the
interval for
which tan 1x =
0 360x° ≤ ≤ °
45 & 225x = ° °
Hint: in what quadrants is
the tangent positive?

17. Example
Write each equation in the
form of an inverse relation:
4
tan
5
α=

18. Example
Write each equation in the
form of an inverse relation:
1
cos
3
β =

19. Try This
Write each equation in the
form of an inverse relation:
sin 1x =
1
sin 1 or arcsin1x x−
= =

20. Example
Find the value of x in the
for which:
cos .6328x =
Leave your answer in
degrees to the nearest tenth.

21. Try This
Find the value of x in the
for which: sin .6328x =
Leave your answer in
degrees to the nearest tenth.
x=39.3
Can you find another value
for x?

22. Definition
The calculator will provide
only the Principal Values of
inverse trig functions:
1
sin x−
1
cos x−
1
tan ( )x−
[ ]90 ,90− ° °
[ ]2, 2π π−
[ ]0,180° [ ]0,πor
or

23. Example
Evaluate the expression.
Assume all angles are in
quadrant 1.
1
sin arcsin
2
 
 ÷
 

24. Example
Evaluate the expression.
Assume all angles are in
quadrant 1.
3
sin arccos
2
 
 ÷ ÷
 

25. Try This
Evaluate the expression.
Assume all angles are in
quadrant 1.
1
os arcsinc
2
 
 ÷
 
3
2

26. Racetrack curves are banked so
that cars can make turns at high
speeds. The proper banking
angle,θ, is given by: 2
tan
v
gr
θ =
where v is the velocity of the car,
g is the acceleration of gravity &
r is the radius of the turn. Find θ
when r=1000ft & v=180 mph.
Example

27. Lesson Close
In order to have an inverse
trig function, we must
restrict the domain so that
duplicate values are
eliminated.