Solving Trigonometric Equations

Key Questions

  • What are other methods for solving equations that can be adapted to solving trigonometric equations?

    Solving concept. To solve a trig equation, transform it into one, or many, basic trig equations. Solving a trig equation, finally, results in solving various basic trig equations.
    There are 4 main basic trig equations:
    sin x = a; cos x = a; tan x = a; cot x = a.
    Exp. Solve sin 2x - 2sin x = 0
    Solution. Transform the equation into 2 basic trig equations:
    2sin x.cos x - 2sin x = 0
    2sin x(cos x - 1) = 0.
    Next, solve the 2 basic equations: sin x = 0, and cos x = 1.
    Transformation process.
    There are 2 main approaches to solve a trig function F(x).
    1. Transform F(x) into a product of many basic trig functions.
    Exp. Solve F(x) = cos x + cos 2x + cos 3x = 0.
    Solution. Use trig identity to transform (cos x + cos 3x):
    F(x) = 2cos 2x.cos x + cos 2x = cos 2x(2cos x + 1 ) = 0.
    Next, solve the 2 basic trig equations.
    2. Transform a trig equation F(x) that has many trig functions as variable, into a equation that has only one variable. The common variables to be chosen are: cos x, sin x, tan x, and tan (x/2)
    Exp Solve sin ^2 x + sin^4 x = cos^2 xsin2x+sin4x=cos2x
    Solution. Call cos x = t, we get
    (1 - t^2)(1 + 1 - t^2) = t^2.
    Next, solve this equation for t.
    Note . There are complicated trig equations that require special transformations.

    Nghi N · 1 · Mar 7 2018
  • How do you find all solutions trigonometric equations?

    As a general description, there are 3 steps. These steps may be very challenging, or even impossible, depending on the equation.

    Step 1: Find the trigonometric values need to be to solve the equation.
    Step 2: Find all 'angles' that give us these values from step 1.
    Step 3: Find the values of the unknown that will result in angles that we got in step 2.

    (Long) Example
    Solve: 2sin(4x- pi/3)=1

    Step 1: The only trig function in this equation is sin.
    Sometimes it is helpful to make things look simpler by replacing, like this:
    Replace sin(4x- pi/3) by the single letter S. Now we need to find S to make 2S=1. Simple! Make S=1/2
    So a solution will need to make sin(4x- pi/3)=1/2

    Step 2: The 'angle' in this equation is (4x- pi/3). For the moment, let's call that theta. We need sin theta = 1/2
    There are infinitely many such theta, we need to find them all.

    Every theta that makes sin theta = 1/2 is coterminal with either pi/6 or with (5 pi)/6. (Go through one period of the graph, or once around the unit circle.)
    So theta Which, remember is our short way of writing 4x- pi/3 must be of the form: theta = pi/6+2 pi k for some integer k or of the form theta = (5 pi)/6 +2 pi k for some integer k.

    Step 3:
    Replacing theta in the last bit of step 2, we see that we need one of: 4x- pi/3 = pi/6+2 pi k for integer k
    or 4x- pi/3 = (5 pi)/6+2 pi k for integer k.

    Adding pi/3 in the form (2 pi)/6 to both sides of these equations gives us:
    4x = (3 pi)/6+2 pi k = pi/2+2 pi k for integer k or
    4x = (7 pi)/6+2 pi k for integer k.

    Dividing by 4 (multiplying by 1/4) gets us to:

    x= pi/8+(2pi k)/4 or
    x=(7 pi)/24+(2 pi k)/4 for integer k.

    We can write this in simpler form:
    x= pi/8+pi/2 k or
    x=(7 pi)/24+pi/2 k for integer k.

    Final note The Integer k could be a positive or negative whole number or 0. If k is negative, we're actually subtracting from the basic solution.

    Jim H · 3 · Mar 9 2015
  • How do you express trigonometric expressions in simplest form?

    Examples of trig expressions: f(x) = sin 2x + cos x;
    f(x) = sin x + sin 2x + sin 3x
    Examples of trig equations: f(x) = sin 2x + cos x = 0
    f(x) = sin x + sin 2x + sin 3x = 0
    Examples of trig inequalities f(x) = sin 2x + cos x > 0
    f(x) = sin x + sin 2x + sin 3x < 0

    Use trig Transformation Identities to transform these above trig expressions into trig basic expressions, or expressions in simplest form.
    Example: Transform f(x) = sin 2x + cos x. Use Identity (sin 2a = 2 sin a*cos a) to transform f(x).
    f(x) = 2*sin x*cos x + cos x = cos x*(2sin x + 1)
    This is f(x) expressed in simplest form.
    Trig equation in simplest form: f(x) = cos x*(sin 2x + 1) = 0
    Trig inequality in simplest form: f(x) = cos x*(2sinx + 1) > 0

    Nghi N. · 2 · Apr 12 2015

Questions

Trigonometric Identities and Equations