Trigonometric equations become more difficult to solve when we are required to use our knowledge of trigonometric identities to manipulate them into a form that we can solve. Factorising becomes a key skill here, allowing us to solve trigonometric equations that involve more than one trigonometric ratio.
Remember: after any algebraic manipulation, we only ever use a positive value in our calculator to find the relative acute angle that solves the equation. We then draw a quadrant (ASTC) diagram to locate the values we need, and finally, check that our answers fit in the domain specified in the question.
In this set, we will consider another five types of trigonometric equations you might need to solve.
Harder types of trigonometric equations in degrees
Type 1: equations that can be factorised
We can never cancel trigonometric functions that appear on both sides of an equation, but we can factorise! Trigonometric equations might have common factors. They can also be factorised as quadratics and the null factor law can be applied to find various solutions. Sometimes, these component factors might not be able to be solved as they involve a value that lies outside the range of sine or cosine, which are the values between $-1$−1 and $1$1 inclusive.
Practice question
Question 1
Find the measure in degrees of the angles satisfying $7\cos^2\left(\theta\right)+3\cos\theta=3$7cos2(θ)+3cosθ=3 for $0^\circ\le\theta\le360^\circ$0°≤θ≤360°. Give your answers correct to one decimal place.
Type 2: equations that can be simplified using trigonometric identities
If an equation has more than one trigonometric function in it and you can't factorise it into a form that you can solve, you will need to manipulate it into a different form using trigonometric identities. The Pythagorean identities are commonly used in these difficult scenarios.
Practice question
Question 2
Solve the equation $2\cos^2\left(\theta\right)=2-\sin\theta$2cos2(θ)=2−sinθ for $0^\circ\le\theta<360^\circ$0°≤θ<360°.
Write your solutions on the same line, separated by commas.
Harder types of trigonometric equations in radians
Solving trigonometric equations in radians uses the same steps as when we solved them using degrees:
1. Find the related positive acute angle (by using exact values or the calculator).
2. Draw a unit circle and use the ASTC rule to determine the relevant quadrants.
3. Determine the value of the angles in the relevant quadrants that satisfy the equation in the given domain.
For harder equations, remember to simplify first. This may involve using trigonometric identities or algebraic techniques like factorising.
Always pay careful attention to the domain in which the angle can lie.
Remember to modify the domain for equations with compound angles.
Practice questions
Question 3
Solve $\tan^2\left(x\right)+2\tan x+1=0$tan2(x)+2tanx+1=0 over the interval $[$[$0$0, $2\pi$2π$)$).