Solve Trigonometric Equations
Examples and Questions With Detailed Solutions

How to solve trigonometric equations? Grade 11 trigonometry questions are presented along with detailed solutions and explanations.

How to Solve Trigonometric Functions?


Example 1

Find all the solutions of the trigonometric equation √3 sec(θ) + 2 = 0
Solution:
Using the identity sec(θ) = 1 / cos(θ), we rewrite the equation in the form
cos(θ) = - √3 / 2
Find the reference θr angle by solving cos(θr) = √3 / 2 for θr acute.
θr = π/6
Use the reference angle θr to determine the solutions θ1 and θ2 on the interval [0 , 2π) of the given equation. The equation cos(θ) = - √3 / 2 suggests that cos(θ) is negative and that means the terminal side of angle θ solution to the given equation is either in quadrants II or III as shown below using the unit circle.

graphical solution of cos(x) = 1/2

Hence the solutions:
θ1 = π - θr = π - π/6 = 5π/6
θ2 = π + θr = π + π/6 = 7π/6
Use the solutions on the interval [0 , 2π) to find all solutions by adding multiples of 2π as follows:
θ1 = 5π/6 + 2nπ , n = 0, ~+mn~ 1 , ~+mn~ 2, ...
θ2 = 7π/6 + 2nπ , n = 0, ~+mn~ 1 , ~+mn~ 2, ...
Below are shown the graphical solutions on the interval [0 , 2π)

graphical solution of cos(x) = 1/2 .



Example 2

Solve the trigonometric equation 2 sin(θ) = -1
Solution:
Rewrite the above equation in simple form as shown below.
sin(θ) = -1/2
Find the reference θr angle by solving sin(θ) = 1/2 for θr acute.
θr = π/6
Use the reference angle θr to determine the solutions θ1 and θ2 on the interval [0 , 2π) of the given equation. The equation sin(θ) = - 1 / 2 suggests that sin(θ) is negative and that means the terminal side of angle θ is either in quadrants III or VI as shown in the unit circle below.

graphical solution of sin(x) = - 1/2

Hence the solutions:
θ1 = π + θr = 7π/6
θ2 = 2π - θr = 11π/6
Use the solutions on the interval [0 , 2π) to find all solutions by adding multiples of 2π as follows:
θ1 = 7π/6 + 2nπ , n = 0, ~+mn~ 1 , ~+mn~ 2, ...
θ2 = 11π/6 + 2nπ , n = 0, ~+mn~ 1 , ~+mn~ 2, ...

graphical solution of cos(x) = 1/2 .



Example 3

Solve the trigonometric equation √2 cos(3x + π/4) = - 1
Solution:
Let θ = 3x + π/4 and rewrite the equation in simple form.
√2 cos(θ) = - 1
cos(θ) = -1/√2
Find the reference θr angle by solving cos(θ) = 1/√2 for θr acute.
θr = π/4
Use the reference angle θr to determine the solutions θ1 and θ2 on the interval [0 , 2π) of the given equation. The equation cos(θ) = - 1/√2 suggests that cos(θ) is negative and that means the terminal side of angle θ is either in quadrants II or III. Hence the two solutions of the equation cos(θ) = - 1/√2 on the interval [0 , 2π) are given by
θ1 = π - θr = 3π/4
θ2 = π + θr = 5π/4
We now write the general solutions by adding multiples of 2π as follows:
θ1 = 3π/4 + 2nπ , n = 0, ~+mn~ 1 , ~+mn~ 2, ...
θ2 = 5π/4 + 2nπ , n = 0, ~+mn~ 1 , ~+mn~ 2, ...
We now substitute θ1 and θ2 by the expression 3x + π/4
3x + π/4 = 3π/4 + 2nπ
3x + π/4 = 5π/4 + 2nπ
and solve for x to obtain the solutions for x.
x = π/6 + 2nπ/3 , n = 0, ~+mn~ 1 , ~+mn~ 2, ...
x = π/3 + 2nπ/3 , n = 0, ~+mn~ 1 , ~+mn~ 2, ...


Example 4

Solve the trigonometric equation - 2 sin 2x - cos x = - 1
Solution:
The above equation may be factored if all trigonometric functions included in that equation are the same. So using the identity sin 2x = 1 - cos 2x, we can rewrite the above equation using the same trigonometric function cos x as follows:
- 2 (1 - cos
2x) - cos x = - 1
Simplify and rewrite as
2 cos
2x - cos x - 1 = 0
Factor the left hand side
(2 cos x + 1)(cos x - 1) = 0
Hence the two equations to solve
(1) 2 cos x + 1 = 0 and (2) cos x - 1 = 0
Solve equation (1) using the reference angle as was done in the examples above.
cos x = -1/2
x
1 = 2π/3 + 2nπ , n = 0, ~+mn~ 1 , ~+mn~ 2, ...
x
2 = 4π/3 + 2nπ , n = 0, ~+mn~ 1 , ~+mn~ 2, ...
Solve equation (2)
cos x = 1
x
3 = 2nπ , n = 0, ~+mn~ 1 , ~+mn~ 2, ...


More References and links

Middle School Maths (Grades 6, 7, 8, 9) - Free Questions and Problems With Answers
High School Maths (Grades 10, 11 and 12) - Free Questions and Problems With Answers
Primary Maths (Grades 4 and 5) with Free Questions and Problems With Answers
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