how to solve equations with trig functions

worked example: Solve $\sin (3x)=\frac{-1}{\sqrt{3}}\cos (3x)$ for $x\in [-\pi ,\pi ]$

worked example: Solve $2{\cos }^2(x)-3\cos (x)+1=0$ for $x\in [-\pi ,2\pi )$

worked example: Solve $2\cos (x-\frac{\pi }{3})+1=0$ for a general solution.

The first step is to isolate the trig function.

$\cos (x-\frac{\pi }{3})=-\frac{1}{2}$

Think of an angle for which the trig function can input it

Since this is cosine, and ASTC, we know that there can be a solution in the second and third quadrants.

second quadrant: $\cos (\pi -\frac{\pi }{3})=-\frac{1}{2}$ so $x=\pi$
third quadrant: $\cos (\pi +\frac{\pi }{3})=-\frac{1}{2}$ so $x=\frac{5\pi }{3}$

Now we found all the solutions throughout the quadrants. $x=\pi \vee x=\frac{5\pi }{3}$

It’s time to make general solutions.

We know we can add as many $2\pi$ as we want to an angle, so therefore we add $+2k\pi$ where $k$ is an integer.

general solution: $x=\pi +2k\pi \vee x=\frac{5\pi }{3}+2k\pi$

worked example: The temperature in a city, $T$ degrees Celsius, at $t$ hours after midnight, is $T(t)=16-7\sin (\frac{\pi }{12}\cdot t)$. what is the temperature at 10:00?
$T(10)=16-7\sin (\frac{\pi }{12}\cdot 10)$
$T(10)=16-7\sin (\frac{5\pi }{6})$
$T(10)=16-7\sin (\frac{\pi }{6})$
$T(10)=16-7\cdot \frac{1}{2}$
$T(10)=12.5$ (degrees Celsius)

worked example: An elevator moves in a straight line. its height above the ground, $x$ meters, at $t$ seconds is $x(t)=2+4\cos (2t)$.

• what is the elevator’s max height?
  $2+4=6$
• at what time for $t>0$ will the elevator first reach the max height?
  we need to find the smallest value of $t$ for $t>0$ that gives $\cos (2t)=1$
  $\cos (0)=1$
  $\cos (\pi )=-1$
  $\cos (2\pi )=1$
  $2t=2\pi$
  $t=\pi$
  the elevator will first reach the max height at $\pi$ seconds

worked example: $h(t)=3\cos (\frac{\pi }{6}\cdot t)$ represents the water height, $h(t)$, meters above sea level, at $t$ hours after midnight. During what time intervals is the water level at least 1.5 meters above sea level?

to answer this question, we want to solve the inequality $3\cos (\frac{\pi }{6}\cdot t)>1.5$

this inequality is best solved by graphing!

amplitude is 3, period is 12

worked example: The equation $3\sin (x)-1=b$ has 1 solution in the interval $x\in (0,2\pi )$. What values can $b$ be?
a solution means a point where $x=0$
as $b$ changes, the value of $x$ changes from the formula $3\sin (x)-1=b$