Inverse Trigonometric Functions — Practice
Sine repeats forever, so its inverse has to be told which answer to give back.
What these questions cover
These questions use 12 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Repeating — Every fraction becomes a decimal that either stops or repeats forever.
- Function — A rule pairing each input with exactly one output.
- Range — The set of outputs that actually occur.
- Adjacent — The side next to that angle which is not the hypotenuse.
- Sine — sin θ = oppositehypotenuse.
- Cosine — cos θ = adjacenthypotenuse.
- Composition — Running one function on the output of another, written (f ° g)(x) = f(g(x)).
- Reference angle — The acute angle between the terminal side and the x-axis.
- Arcsine — The inverse of sine, restricted to outputs from -90° to 90°.
- Arccosine — The inverse of cosine, restricted to outputs from 0° to 180°.
- Arctangent — The inverse of tangent, restricted to outputs strictly between -90° and 90°.
- Quadrant — One of the four regions the axes divide the plane into.
Problem 1
Find arcsin12 in degrees.
Worked answer
30
sin 30° = 12
Problem 2
Find arccos 0 in degrees.
Worked answer
90
cos 90° = 0
Problem 3
Find arctan 1 in degrees.
Worked answer
45
tan 45° = 1
Problem 4
Find arcsin 0 in degrees.
Worked answer
0
sin 0° = 0
Problem 5
Find arccos 1 in degrees.
Worked answer
0
cos 0° = 1
Problem 6
Find arcsin(-12) in degrees.
Worked answer
-30
Reference angle 30°
Arcsine returns -90° to 90°, so -30°
Problem 7
Find arccos(-12) in degrees.
Worked answer
120
Reference angle 60°
Cosine is negative in quadrant II, so 180-60=120
Problem 8
Find arctan(-1) in degrees.
Worked answer
-45
Reference angle 45°
Negative, so -45°
Problem 9
Find arcsin√32 in degrees.
Worked answer
60
sin 60° = √32
Problem 10
Find arccos√22 in degrees.
Worked answer
45
cos 45° = √22
Problem 11
What is the range of arccos?
- 0° to 180°
- -90° to 90°
- 0° to 90°
- -180° to 180°
Worked answer
0° to 180°
Cosine falls from 1 to -1 across 0° to 180°
That covers its whole range exactly once
Problem 12
What is the range of arcsin?
- -90° to 90°
- 0° to 180°
- 0° to 360°
- -180° to 0°
Worked answer
-90° to 90°
From -90° to 90° sine rises from -1 to 1 without repeating
Problem 13
Evaluate arcsin(sin 150°) in degrees.
Worked answer
30
sin 150° = 12
The angle in [-90°, 90°] with that sine is 30°
Problem 14
How many values does arcsin 2 have?
- None
- One
- Two
- Infinitely many
Worked answer
None
Sine never exceeds 1
So no angle has sine 2
Problem 15
Evaluate sin(arcsin 0.4).
Worked answer
0.4
0.4 is inside [-1,1], so the composition returns it unchanged
Problem 16
An obtuse angle has sin A = 0.8. Find A to the nearest degree.
Worked answer
127
sin-1(0.8) ≈ 53°
Obtuse: 180-53 = 127°