Question

How would you solve the following trig equation sin(x)=-0.5 by isolating the x?

Answer 1

To get x by itself in a trig equation like this, you use the inverse sine. It's written on your calculator as sin⁻¹ and often in books as arcsin to differentiate it from the negative exponent. Taking the sine and inverse sin of any angle brings you back to the original angle.

sin (x) = -0.5

arcsin (sin(x)) = arcsin (-0.5)

x = arcsin (-0.5)

Since you are asking "what angle has its sine as -1/2?" this problem can be answered with a 30- 60 - 90 triangle or with a calculator. In degree mode, you would this be -30 degrees, or 330 degrees and 210 degrees, as the sine function is negative in quadrants III and IV.

Answer 2

Make use of the inverse sine function. Take the inverse sine of both sides of the equation. Of course, within the appropriate limits, the inverse sine of the sine function is the original argument, as is the case with any inverse function: f⁻¹(f(x)) = x.

... sin⁻¹(sin(x)) = sin⁻¹(-0.5)

... x = sin⁻¹(-0.5)

... x = -30°

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You need to be careful with inverses of trig functions, because they are only defined over a limited domain and range. The range of the inverse sine function is -90° to 90°, so, for example, sin⁻¹(sin(150°)) = sin⁻¹(0.5) = 30°.