5.3.4 Strategy for Trigonometric Equations

Solving trigonometric equations can be challenging, but with a systematic approach, you can tackle them efficiently. Here's a general strategy to help you solve a wide range of trigonometric equations:

1. Understand the Equation

Identify which trigonometric function(s) are involved (e.g., $\sin \theta, \cos \theta, \tan \theta$ ).
Determine whether the equation is linear (e.g., $\sin \theta = \frac{1}{2}$ ), quadratic (e.g., $2\sin^2 \theta - \sin \theta - 1 = 0$ ), or involves multiple trigonometric functions.

2. Simplify the Equation

Isolate the trigonometric function: Get the trigonometric function (e.g., $\sin \theta, \cos \theta$ ) by itself on one side of the equation.
Factor or Simplify: If the equation is quadratic or more complex, try factoring, expanding, or simplifying using trigonometric identities.
Substitution: If multiple trigonometric functions are involved (e.g., $\sin^2 \theta + \cos^2 \theta = 1$ ), consider substituting using identities to reduce the equation to one trigonometric function.

3. Use Trigonometric Identities

Apply identities such as the Pythagorean identities, double-angle identities, sum-to-product identities, or half-angle identities to simplify the equation.
Common identities include: $\sin^2 \theta + \cos^2 \theta = 1$ $\tan \theta = \frac{\sin \theta}{\cos \theta}$ $\sin(2\theta) = 2\sin \theta \cos \theta$

4. Solve the Simplified Equation

Linear Equations:
- Solve for the angle by taking the inverse trigonometric function (e.g., $\sin^{-1}, \cos^{-1}, \tan^{-1}$ ).
- Consider all solutions within the given interval by accounting for the periodic nature of the trigonometric functions.
Quadratic Equations:
- Factor if possible, or use the quadratic formula: $u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
- Back-substitute to solve for the angle.
- Check for extraneous solutions, especially if you square both sides of the equation.

5. Consider the General Solution

Trigonometric functions are periodic, meaning they repeat their values at regular intervals.
For $\sin \theta = k and \cos \theta = k:$ $sin θ = kan d cos θ = k :$
- $\theta = \theta_0 + 360^\circ n\ (or\ 2\pi$ n) where n is any integer.
- Include both primary solutions $\theta_0 and 180^\circ - \theta_0 (or\ \pi - \theta_0).$
For $\tan \theta = k:$ $tan θ = k :$
- $\theta = \theta_0 + 180^\circ n (or\ \pi n).$

6. List All Solutions in the Given Interval

Ensure that all solutions are within the specified interval, typically $0^\circ to\ 360^\circ$ or $0\ to\ 2\pi$ radians.
Adjust the general solution accordingly to fit within this range.

7. Verify Solutions

Substitute each solution back into the original equation to ensure it satisfies the equation.
Discard any extraneous solutions that might have arisen from squaring both sides or other manipulations.

infoNote

Example Strategy Application:

Problem: Solve $2\sin^2 \theta - 3\sin \theta + 1 = 0 for 0^\circ \leq \theta \leq 360^\circ.$

Identify: This is a quadratic trigonometric equation in $\sin \theta.$
Substitute: Let u = $\sin \theta$ , so the equation becomes $2u^2 - 3u + 1 = 0.$
Factor: Factor the quadratic: $(2u - 1)(u - 1) = 0$ So, u = $\frac{1}{2}\ or\ u = 1.$
Back-substitute: Replace $u$ with $\sin \theta:$

$\sin \theta = \frac{1}{2}$
$\sin \theta = 1$

Solve:

For $\sin \theta = \frac{1}{2}:$ $\theta = 30^\circ, 150^\circ$
For $\sin \theta = 1:$ $\theta = 90^\circ$

List All Solutions: The solutions in the interval $0^\circ \leq \theta \leq 360^\circ$ are: $\theta = 30^\circ, 90^\circ, 150^\circ$
Verify: Substitute these angles back into the original equation to ensure they are correct.

Summary:

Simplify the trigonometric equation using identities or substitution.
Solve the resulting equation for the variable.
Consider the periodic nature of trigonometric functions to find all solutions within the given interval.
Verify the solutions to ensure correctness.

Solving Trig Equations Involving Compound Angles

By "compound angle," we mean "an angle more complicated than just $\theta.$ "

infoNote

Example: Solve $\sin(3\theta) = 0.42, 0 \leq \theta \leq 180$ .
Notice we are solving for $\theta$ , but the angle is $3$ $\theta$ within the $\sin$ function.
$0 \leq 3\theta \leq 540 ( \times 3 )$

Modify the domain to find limits for the compound angle in the bracket.
Find all solutions for our compound angle (in this case, it's $3\theta$ )

$3\theta = \arcsin(0.42) \approx 24.85^\circ$

Calculator Steps:
Press [ $\sin^{-1}$ ] then the letters to recall this number.
Store long numbers by pressing $[ STO ]$ then a letter with a red letter.
Solutions: $3\theta = :highlight[24.85^\circ, 155.15^\circ, 384.834^\circ, 515.165^\circ]$

Find $\theta$ for each intermediate solution (in this case, divide all by $3$ ):

$\theta = :success[8.278^\circ, 51.72^\circ, 128.3^\circ, 171.7^\circ] (4sf)$

infoNote

Example: Solve $\cos(x - 37.6^\circ) = 0.17, 0 \leq x \leq 540$ .

$\boxed {x = :success[137.4^\circ, 297.8^\circ, 497.4^\circ] \quad \text{(4sf)}}$

Strategy for Trigonometric Equations Simplified Revision Notes for A-Level AQA Maths Pure

5.3.4 Strategy for Trigonometric Equations

1. Understand the Equation

2. Simplify the Equation

3. Use Trigonometric Identities

4. Solve the Simplified Equation

5. Consider the General Solution

6. List All Solutions in the Given Interval

7. Verify Solutions

Example Strategy Application:

Summary:

Solving Trig Equations Involving Compound Angles

500K+ Students Use These Powerful Tools to Master Strategy for Trigonometric Equations For their A-Level Exams.

Other Revision Notes related to Strategy for Trigonometric Equations you should explore