Notes
Solving on an interval
- Find the reference angle, then use the quadrants where the function has the needed sign (sine: I, II; cosine: I, IV; tangent: I, III for positive values).
- For $\sin kx = a$ on $[0, 2\pi)$, list all values of $kx$ in $[0, 2\pi k)$, then divide by $k$.
General solutions
- $\sin x = a$: $x = \alpha + 2\pi n$ or $\pi - \alpha + 2\pi n$. $\cos x = a$: $x = \pm\alpha + 2\pi n$. $\tan x = a$: $x = \alpha + \pi n$.
Algebra and identities
- Factor and set each factor to 0. Do not divide by a trig function that can be 0.
- Use identities such as $\sin^2 x = 1 - \cos^2 x$ or $\sin 2x = 2\sin x\cos x$ to get one function.
- Values outside $[-1, 1]$ are impossible for sine and cosine.
Topics in this unit
Each topic has its own short notes and practice.
Practice
Sine is negative in Quadrants III and IV, with reference angle $\displaystyle \frac{\pi}{3}$.
Sine is positive in Quadrants I and II, with reference angle $\displaystyle \frac{\pi}{4}$.
$\displaystyle \cos x = -\frac{\sqrt{3}}{2}$, which happens in Quadrants II and III with reference angle $\displaystyle \frac{\pi}{6}$.
$2x = \pi$ or $3\pi$ (in $[0, 4\pi)$), so $\displaystyle x = \frac{\pi}{2}$ or $\displaystyle \frac{3\pi}{2}$.
$\sin x(\sin x - 1) = 0$, so $\sin x = 0$ (giving $0$ and $\pi$) or $\sin x = 1$ (giving $\displaystyle \frac{\pi}{2}$).
Tangent is negative in Quadrants II and IV, with reference angle $\displaystyle \frac{\pi}{4}$.
Free response
Write your answer first, then open the scoring guide.
Free-response question
Consider the equation $2\cos^2 x + \cos x - 1 = 0$.
(a) Factor the left side.
Show the answer and scoring guide
$2\cos^2 x + \cos x - 1 = (2\cos x - 1)(\cos x + 1)$.
Check off the points you earned:
(b) Solve the equation on the interval $[0, 2\pi)$.
Show the answer and scoring guide
$\displaystyle \cos x = \frac{1}{2}$ gives $\displaystyle x = \frac{\pi}{3}$ or $\displaystyle \frac{5\pi}{3}$. $\cos x = -1$ gives $x = \pi$. The solutions are $\displaystyle \frac{\pi}{3}$, $\pi$ and $\displaystyle \frac{5\pi}{3}$.
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(c) Write the general solution for all real numbers.
Show the answer and scoring guide
$\displaystyle x = \pm\frac{\pi}{3} + 2\pi n$ or $x = \pi + 2\pi n$, where $n$ is an integer.
Check off the points you earned: