Trigonometric equations review

Review your trigonometric equation skills by solving a sequence of equations in increasing complexity.

Practice set 1: Basic equations

Example: Solving sin(x)=0.55\sin(x)=0.55

Let's use the calculator and round to the nearest hundredth.
sin1(0.55)=0.58\sin^{-1}(0.55)=0.58
(We are using radians.)
We can use the identity sin(πθ)=sin(θ)\sin(\pi-\theta)=\sin(\theta) to find the second solution within [0,2π][0,2\pi].
π0.58=2.56\pi-0.58=2.56
We use the identity sin(θ+2π)=sin(θ)\sin(\theta+2\pi)=\sin(\theta) to extend the two solutions we found to all solutions.
x=0.58+n2πx=0.58+n\cdot2\pi
x=2.56+n2πx=2.56+n\cdot2\pi
Here, nn is any integer.

Check your understanding

Problem 1.1
Select one or more expressions that together represent all solutions to the equation.
The answers are in radians. nn is any integer.
cos(x)=0.15\cos(x)=0.15
Choose all answers that apply:
Choose all answers that apply:

Want to try more problems like this? Check out this exercise.

Practice set 2: Advanced equations

Example: Solving 16cos(15x)+8=216\cos(15x)+8=2

First, we isolate the trig expression:
16cos(15x)+8=216cos(15x)=6cos(15x)=0.375\begin{aligned}16\cos(15x)+8&=2\\\\ 16\cos(15x)&=-6\\\\ \cos(15x)&=-0.375\end{aligned}
Use the calculator and round to the nearest thousandth:
cos1(0.375)=1.955\cos^{-1}(-0.375)=1.955
Use the identity cos(θ)=cos(θ)\cos(\theta)=\cos(-\theta) to find that the second solution within [π,π][-\pi,\pi] is 1.955-1.955.
Use the identity cos(θ)=cos(θ+2π)\cos(\theta)=\cos(\theta+2\pi) to find all the solutions to our equation from the two angles we found above. Then we solve for xx (remember that our argument is 15x15x):
15x=1.955+n2πx=1.955+n2π15x=0.130+n2π15\begin{aligned} 15x&=1.955+n\cdot2\pi \\\\ x&=\dfrac{1.955+n\cdot2\pi}{15} \\\\ x&=0.130+n\cdot\dfrac{2\pi}{15} \end{aligned}
Similarly, the second solution is x=0.130+n2π15x=-0.130+n\cdot\dfrac{2\pi}{15} .

Check your understanding

Problem 2.1
Select one or more expressions that together represent all solutions to the equation.
The answers are in radians. nn is any integer.
20sin(10x)10=520\sin(10x)-10=5
Choose all answers that apply:
Choose all answers that apply:

Want to try more problems like this? Check out this exercise.

Practice set 3: Word problems

Problem 3.1
L(t)L(t) models the length of each day (in minutes) in Manila, Philippines tt days after the spring equinox. Here, tt is entered in radians.
L(t)=52sin(2π365t)+728L(t) = {52}\sin\left({\dfrac{2\pi}{365}}t\right) + {728}
What is the first day after the spring equinox that the day length is 750750 minutes?
Round your final answer to the nearest whole day.
  • Your answer should be
  • an integer, like 66
  • a simplified proper fraction, like 3/53/5
  • a simplified improper fraction, like 7/47/4
  • a mixed number, like 1 3/41\ 3/4
  • an exact decimal, like 0.750.75
  • a multiple of pi, like 12 pi12\ \text{pi} or 2/3 pi2/3\ \text{pi}
days

Want to try more problems like this? Check out this exercise.
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