Download Section 6.2: Solving Trigonometric Equations I

Chapter 6
Section 6.2
Section 6.2: Solving Trigonometric Equations I
I.
Hints for Solving a Trigonometric Equation
1. Decide whether the equation is linear or quadratic in form, so you can determine the solution
method.
2. If only one trigonometric function is present, first solve the equation for that function.
3. If more than one trigonometric function is present, rearrange the equation so that one side
equals 0. Then try to factor and set each factor equal to 0 to solve.
4. If the equation is quadratic in form, but not factorable, use the quadratic formula. Check that
solutions are in the desired interval.
5. Try using identities to change the form of the equation. It may be helpful to square each side of
the equation first. In this case, check for extraneous solutions.
II.
Solving by Linear Methods
Example 1: Solve 3tan  3  0
a) over the interval 0,360
b) for all solutions.

Chapter 6
Section 6.2
Practice: Solve the equation 2sec x  1  sec x  3 :
a) over the interval  0, 2 
b) for all solutions.
III.
Solving by Factoring

Just as some algebraic equations can be factored into two or more linear equations, some
trigonometric equations can also be factored.

Caution: Trying to solve an equation in by dividing each side by a variable expression can
easily lead to lost solutions. For this reason, avoid dividing by a variable expression.
Example 2: Solve the equation cos cot    cos over the interval 0,360

Chapter 6
Section 6.2
Practice: Solve sin x tan x  sin x over the interval  0, 2  .
IV.
Solving by Quadratic Methods

Remember from algebra that an equation of the form au 2  bu  c  0 , where u is an algebraic
expression, is called quadratic in form. Such equations can be solved by either:
1. Factoring
2. Quadratic Formula
√
Example 3 (Factoring): Solve 3sin 2 x  sin x  2  0 over the interval  0, 2  .
Chapter 6
Section 6.2
Practice: Solve the equation 2cos2 x  cos x  1 over the interval 0,360  .
Example 4 (Quadratic Formula): Find all solutions of cos x  cos x  2   1 , where x is in radians.
Chapter 6
V.
Section 6.2
Using Trigonometric Identities

One major use of trigonometric identities is they allow you to change the form of an equation
to a form that is more easily solvable.
Example 5: Solve cot x  3  csc x over the interval  0, 2  .
Practice: Find all solutions of the following equations:
a) cos2 x  sin 2 x  0
b) sec2 x  2 tan x  4