When $\cos \theta = \tan \theta$, find the exact value of $\sin \theta$
*Write your answer in the form $\frac{a+\sqrt{b}}{c}$ where $a, b$ and $c$ are integers.
Solve for $x$ on the domain $0 \le x \lt 2\pi$.
a) $\sin{x}+\cos{x}=1$
b) $\sin{x}+\sin^2{x}=\cos^2{x}$
c) $2\sin^2{x}+(4-\sqrt{3})\cos{x} -2(1-\sqrt{3})=0$
d) $2\sin^2{\left(\frac{1}{2}x+\frac{\pi }{4}\right)}-\sin{\left(\frac{1}{2}x+\frac{\pi }{4}\right)}-1=0$
e) $\displaystyle\frac{\sin{x}}{1-\cos{x}} – \frac{\sin{x}}{1+\cos{x}} =2$
f) $3\sin^2{x}+2\sqrt{3}\sin{x} \cos{x}-3\cos^2{x}=0$
*For multiple answers, separate with commas. If your answer contains $\pi$, write in the form $\frac{\pi}{b}$ or $\frac{a\pi}{b}$ where $a$ and $b$ are integers. For example: $\frac{\pi}{3}$ or $\frac{5\pi}{6}$ or $2\pi$
a)
b)
c)
d)
e)
f)
For the function $f\left(x\right)=\sin{x}+\cos^2{x}+1$, find the coordinates of the first maximum point and minimum point for $x \ge 0$
Maximum point:
Minimum point:
Find the four values of $\theta$ for $0 \le \theta \lt 2\pi$ that satisfy $\displaystyle\frac{1}{\sin^2\theta}-\frac{1}{\cos^2\theta}-\frac{1}{\tan^2\theta}-\frac{1}{\csc^2\theta}-\frac{1}{\sec^2\theta}-\frac{1}{\cot^2\theta}=-3$
*Separate your answers with commas. If your answer contains $\pi$, write in the form $\frac{\pi}{b}$ or $\frac{a\pi}{b}$ where $a$ and $b$ are integers. For example: $\frac{\pi}{3}$ or $\frac{5\pi}{6}$ or $2\pi$
Find the range of the function $y=\sin^4 x + \cos^2 x$
$\le y \le$
Given $\tan \theta = \frac{3}{4}$, find the value of $\sin 2 \theta$ for $\pi \lt \theta \lt \frac{3}{2} \pi$
*Give your answer as a fraction in simplest terms.
Acute angles $A,B$ and $C$ satisfy the following system of equations:
$$\begin{cases}
\cos A=\tan B \\
\cos B=\tan C \\
\cos C=\tan A
\end{cases}$$
Find the exact value of $\sin^2 A$.
Given that $a \sin{4x}=b \sin{2x}$ and $0 \lt x \lt \frac{\pi}{2}$, express $\sin^2{x}$ in terms of $a$ and $b$.
*Express your answer as a single fraction in simplest terms
In $\Delta ABC$, if the length of $BC$ is twice the length of $AC$, and $\angle A-\angle B=90^\circ$, what is the value of $\tan C$?
*Give your answer as a single fraction in simplest form