sin^-1(sin5pi/6)
To find the value of sin^-1(sin5π/6), we can use the concept of inverse trigonometric functions.
1. First, let's rewrite the given expression as sin^-1(sin(150°)).
Note that 5π/6 radians is equivalent to 150°.
2. The inverse sine function (sin^-1) undoes the effects of the sine function (sin), so sin^-1(sin(150°)) will give us the angle whose sine is 150°.
3. However, the range of the inverse sine function is restricted to [-π/2, π/2], which corresponds to [-90°, 90°].
4. Since 150° is outside the range of the inverse sine function, we need to find an equivalent angle within this range.
5. We can make use of the periodic nature of the sine function. A sine wave repeats itself every 360° (or 2π radians). So we can subtract or add multiples of 360° to our angle and still get the same sine value.
6. In this case, we can subtract 360° from 150° until we get an angle within the desired range: 150° - 360° = -210°.
7. Now, we can apply the inverse sine function: sin^-1(sin(-210°)).
8. Note that sin(-210°) = sin(150°) because the sine function is an odd function, meaning sin(-x) = -sin(x).
9. Since 150° is within the range of the inverse sine function, we can evaluate sin^-1(sin(150°)) directly.
10. Therefore, sin^-1(sin5π/6) = -210°.