Honors Geometry
Advanced Trigonometry
6. Inverse Trigonometric Functions
The equation y = sin 30 asks the question “What is the sine of a 30 angle?”
This question has a single answer, ½.
The equation ½ = sin y asks the question “What angle has a sine equal to ½ ?”
This questions has more than one answer : 30, 150, 390, 510, -330, -210, …
The first question was in the form
y = sin x
The second question can be expressed as x = sin y
But we like equations in terms of y.
So we write y = sin-1 x which is read “y equals the inverse sine of x”
(That raised –1 is not an exponent.)
Find y if y = sin-1 x and x = ½.
y could be 30, 150, 390, 510, -330, -210, …, since the sine of each is ½ .
This represents a problem, since there should be only one answer for y.
Even if we say 0 < y < 360 (the angles we usually use),
we still have two possibilities for y, 30and 150.
This means that
(1) sin-1 x is not a function and
(2) we don’t know whether to use 30 or 150.
We solve these two problems with one solution.
We say that, for y = sin-1 x, the range is -90 < y < 90, essentially Quad I and Quad IV.
We need a quadrant where sine is positive (I) and one where sine is negative (IV).
We will also use Quad I and Quad IV for tan-1 x, since tangent is positive in I and
negative in II. So for y = tan-1 x, -90 < y < 90.
Quad I and Quad IV won’t work for cos-1 x, since cosine is positive in both. We use I
(cosine positive) and II (cosine negative). So for y = cos-1 x, 0 < y < 180.
Summary
Trig Functions
y = sin x
y = cos x
y = tan x
Domain
R
R
R
Range
-1 ≤ y ≤ 1
-1 ≤ y ≤ 1
R
Inverse Trig Functions
y = sin-1 x
y = cos-1 x
y = tan-1 x
Domain
-1 ≤ x ≤ 1
-1 ≤ x ≤ 1
R
Range
-90 < y < 90
0 < y < 180
-90 < y < 90
Honors Geometry
Advanced Trigonometry
6. Inverse Trigonometric Functions
Exercises.
Using your calculator, find the measure of the angle whose inverse trig function is give.
Round all angles to the nearest tenth of a degree.
1. sin-1(0.3452) =
25. sin-1( ½ ) =
2. cos-1(0.7689) =
26. cos-1( ½ ) =
3. tan-1(2.345) =
27. tan-1( ½ ) =
4. sin-1(-0.5337) =
28. sin-1( -½) =
5. cos-1(-0.1298) =
29. cos-1(-½) =
6. tan-1(-0.647) =
30. tan-1(-½) =
7. sin-1(1) =
31. sin-1(0.7071) =
8. cos-1(1) =
32. cos-1(0.7071) =
9. tan-1(1) =
33. tan-1(0.7071) =
10. sin-1(0) =
34. sin-1(0.8660) =
11. cos-1(0) =
35. cos-1(0.8660) =
12. tan-1(0) =
36. tan-1(0.8660) =
13. sin-1(-1) =
37. sin-1(-1.732) =
14. cos-1(-1) =
38. cos-1(-1.732) =
15. tan-1(-1) =
39. tan-1(-1.732) =
16. sin-1(2) =
40. sin-1(0.9009) =
17. cos-1(2) =
41. cos-1(-0.2364) =
18. tan-1(2) =
42. tan-1(100) =
19. sin-1(0.2034) =
43. sin-1(-0.3572) =
20. cos-1(0.4553) =
44. cos-1(0.3423) =
21. tan-1(12) =
45. tan-1(-1.435) =
22. sin-1(-0.0056)
46. sin-1(12/13) =
23. cos-1(-0.0328
47. cos-1(-6/11) =
24. tan-1(-0.0364)
48. tan-1(35/8) =