Inverse Sine: Step-by-Step Guide
Arcsine, written arcsin(x) or sin^-1(x), is the inverse of sine. It answers the question 'what angle has this sine?' Because many angles have the same sine, we only allow answers between -pi/2 and pi/2. The input x must be between -1 and 1, and the output is an angle between -pi/2 and pi/2.
Math Racer skill guide · Trigonometry · Grades 11-12
How to solve it, step by step
- Check the input is between -1 and 1. If x is bigger than 1 or less than -1, arcsin(x) does not exist for real numbers.\[-1 \leq x \leq 1\]
- Recognize the value. Compare x to the common sine values: 0, 1/2, root 2 / 2, root 3 / 2, 1, and their negatives.\[\sin\theta \in \{0, \tfrac{1}{2}, \tfrac{\sqrt{2}}{2}, \tfrac{\sqrt{3}}{2}, 1\}\]
- Find the angle between -pi/2 and pi/2. Pick the one angle in that range with the sine you want. Negative inputs give negative angles.\[\theta = \arcsin(x),\;\; \theta \in [-\tfrac{\pi}{2}, \tfrac{\pi}{2}]\]
- Check. Make sure sin(angle) = x and the angle is in the allowed range.\[\sin(\arcsin(x)) = x\]
Worked example
Find arcsin(-sqrt(3)/2).
Step 1: The size of -root 3 / 2 is less than 1, so it is a valid input.
Step 2: We know sin(pi/3) = root 3 / 2.
Step 3: We need a negative angle between -pi/2 and pi/2 with sine -root 3 / 2. Since sine is odd, the angle is -pi/3.
Step 4: Check: sin(-pi/3) = -root 3 / 2, and -pi/3 is between -pi/2 and pi/2. It works.
The answer is arcsin(-root 3 / 2) = -pi/3.
Tips and common mistakes
- Arcsin only gives angles between -pi/2 and pi/2. Never give an answer outside that range.
- Arcsin is odd: arcsin(-x) = -arcsin(x).
- If the input is outside -1 to 1, there is no answer.
- Common values: arcsin(0) = 0, arcsin(1/2) = pi/6, arcsin(root 2 / 2) = pi/4, arcsin(root 3 / 2) = pi/3, arcsin(1) = pi/2.
- sin(arcsin(x)) = x always works for x between -1 and 1. But arcsin(sin(theta)) only equals theta if theta is between -pi/2 and pi/2.
- In calculus, the derivative of arcsin(x) is 1 / the square root of (1 - x^2).
- If your calculator gives degrees, multiply by pi/180 to get radians.
- To solve sin(theta) = k for every angle, arcsin only gives one answer. The other is pi - arcsin(k). Then add 2 pi n to each.