How to Solve for a Variable in a Trig Function

By Karl Wallulis
BananaStock/BananaStock/Getty Images

Trig functions are equations containing the trigonometric operators sine, cosine and tangent, or their reciprocals cosecant, secant and tangent. The solutions to trigonometric functions are the degree values that make the equation true. For example, the equation sin x + 1 = cos x has the solution x = 0 degrees because sin x = 0 and cos x = 1. Use trig identities to rewrite the equation so that there's only one trig operator, then solve for the variable using inverse trig operators.

Rewrite the equation using trigonometic identities, such as the half-angle and double-angle identities, the Pythagorean identity and the sum and difference formulas so that there's only one instance of the variable in the equation. This is the most difficult step in solving trig functions, because it's often unclear which identity or formula to use. For example, in the equation sin x cos x = 1/4, use the double angle formula cos 2x = 2 sin x cos x to substitute 1/2 cos 2x in the left side of the equation, yielding the equation 1/2 cos 2x = 1/4.

Isolate the term containing the variable by subtracting constants and dividing coefficients of the variable term on both sides of the equation. In the above example, isolate the term "cos 2x" by dividing both sides of the equation by 1/2. This is the same as multiplying by 2, so the equation becomes cos 2x = 1/2.

Take the corresponding inverse trigonometric operator of both sides of the equation to isolate the variable. The trig operator in the example is cosine, so isolate the x by taking the arccos of both sides of the equation: arrccos 2x = arccos 1/2, or 2x = arccos 1/2.

Calculate the inverse trigonometric function on the right side of the equation. In the above example, arccos 1/2 = 60 degress or pi / 3 radians, so the equation becomes 2x = 60.

Isolate the x in the equation using the same methods as in Step 2. In the above example, divide both sides of the equation by 2 to get the equation x = 30 degrees or pi / 6 radians.

About the Author

Karl Wallulis has been writing since 2010. He has written for the Guide to Online Schools website, covering academic and professional topics for young adults looking at higher-education opportunities. Wallulis holds a Bachelor of Arts in psychology from Whitman College.