Solve Trigonometric Equations

Tutorial, with detailled explanations, on how to solve trigonometric equations using different methods and strategies and the properties of trigonometric functions and identities. The unit circle helps in locating the solutions once you have the reference angle.

Example 1: Solve the trigonometric equation.(find all solutions)

2 cos x + 2 = 3

Solution to example 1

  • solve for cos(x)

    cos x = 1/2

  • solve for x by finding all values in the interval [0 , 2pi) that statisfy the above trigonometric equation. In this case, with cosine positive and equal to 1 / 2, there are two values: one in the first quadrant of the unit circle.

    x1 = pi / 3

  • and a second one in the fourth quadrant (see the two solutions in unit circle in figure below).

    x2 = 2*pi - pi / 3 = 5*pi / 3

    solutions in unit circle 1

  • find all solutions using the fact that of cos x has a period of 2pi

    x1 = pi / 3 + 2*k*pi

    x2 = 5*pi / 3 + 2*k*pi

    where k is any integer



conclusion: There is an infinite number of solutions which can be generated by giving different values to k.



Example 2: Find the solutions in the interval [0 , 2pi) for the trigonometric equation

-5 cos 2x + 9 sin x = -3

Solution to example 2

  • change cos 2x to 1 - sin 2x

    -5(1 - sin 2x) + 9 sin x = -3

  • mutliply factors and group to obtain

    5 sin2x + 9sin x -2 = 0

  • let u = sinx and substitute to obtain an quadratic equation.


    5 u2 + 9 u - 2 = 0

  • use any method to solve for u. By the quadratic formula, we obtain two solutions u1 and u2

    u1 = [ -9 - sqrt(121) ] / 10 = 1 / 5 = -2

  • and

    u2 = [ -9 + sqrt(121) ] / 10 = 0.2

  • we now solve the equation for x

    u1 = sin x = -2

  • the above equation has no solutions for x since -2 is not in the range of values of sin(x);
    -1 <= sin x <= 1.


    u2 = sin x = 0.2

  • sin x is positive in the first and second quadrants of the unit circle. There are two solutions to the above equation in [0 , 2pi)(see unit circle below)
    solutions in unit circle 2


    solution in the first quadrant
    x1 = arcsin 0.2

  • arcsin 0.2 is the inverse sine function and it can be approximated using a calculator)

    solution in the second quadrant
    x2 = pi - arcsin 0.2
    conclusion: There are two solutions to the given equation.

Example 3: Find all solutions for the trigonometric equation

cot x cos 2x = cot x

Solution to example 3

  • subtract cot x from both sides of the equation and simplify

    cot x cos 2x - cot x = 0

  • Factor cot x

    cot x (cos 2x - 1) = 0

  • Setting each factor in the above trigonometric equation to zero, we obtain two equations.


    cot x = 0 and cos 2x - 1 = 0

  • The solutions to equation cot x = 0 are given by

    x = pi / 2 + k*pi , k is am integer.

  • Equation cos 2x - 1 = 0 gives

    cos x = 1 and cos x = -1

  • The solutions to the above equations are given by

    x = 2k*pi and x = (2k + 1)*pi where k is an integer

  • HOWEVER the above cannot be solution to the given equation since cot x is undefined for x = 2k*pi and x = (2k + 1)*pi.

    conclusion: The solutions to the given equation are.

    x = pi / 2 + k*pi where k is an integer.

Note that many of the techniques used in solving algebraic equations are also used to solve trigonometric equations.


More trigonometric equations to solve, go here


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Updated: 2 April 2013

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