7.10 Solving equations using 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 1
If $\cos\theta=0.9063$cosθ=0.9063, find the value of $\theta$θ correct to two decimal places, where $0$0 ≤
Find the acute angle $\theta$θ that solves the equation.
Now find ALL solutions to $\theta$θ in the range $0$0 to $2\pi$2π (separate alternative solutions by a comma). Round your solutions to $2$2 decimal places.
Question 2
Solve $\tan x=\sqrt{3}$tanx=√3 for $-\pi
Question 3
Solve $4\sin^2\left(x\right)=1$4sin2(x)=1 for $-4\pi
Question 4
Solve $\tan^2\left(x\right)+2\tan x+1=0$tan2(x)+2tanx+1=0 over the interval $[$[$0$0, $2\pi$2π$)$).
Question 5
Solve $\sin\left(x+\frac{\pi}{3}\right)=\frac{1}{2}$sin(x+π3)=12 where $0\le x\le2\pi$0≤x≤2π.
Question 6
Solve $\sin x=0$sinx=0 over the interval $[$[$-4\pi$−4π, $4\pi$4π$]$].
Write your answers on the same line, separated by commas.