SL 3.5—Unit circle definitions of sin, cos, tan. Exact trig ratios, ambiguous case of sine rule Questionbank

For the function $f(x)$ find the amplitude of the function.

Determine the value of $k$ such that the $f(x)$ has a period of $12\pi$

Find the $x$-coordinates of the minima and maxima of the function $f(x)$ within a period of $12\pi$ and $k>0$ under the domain $x\in[0,2\pi]$

Find the derivative of the function $g(x)$.

Find the equation of the tangent line to the graph of $g(x)$ at $x = \frac{\pi}{4}$.

Hence, or otherwise, show that at $x=\frac{\pi}{4}$ the function attains its maximum.

Find an expression for $r$ in terms of $\alpha$.

Find the values of $\alpha$ which give the greatest value of the sum.

Find the common ratio $r$ of this geometric sequence in terms of $\cos$

Given that $|r| \not= 1$, explain why the sum to infinity exists and find it for $x=\frac{\pi}{6}$ giving your answer in the form $\frac{a+b\sqrt{c}}{d}$

$\sin 340^{\circ}$

$\cos 340^{\circ}$

$\tan 340^{\circ}$

$\sin \left(-20^{\circ}\right)$

$\cos \left(-20^{\circ}\right)$

$\tan \left(-20^{\circ}\right)$

Start from the left-hand side and express everything in terms of $\sin\theta$ and $\cos\theta$.

Hence, or otherwise, show that $\sin(2\theta)=\frac{2\tan\theta}{1+\tan^2\theta}.$

If $\tan\theta=\dfrac{3}{4}$, find the exact values of $\sin(2\theta)$ and $\cos(2\theta)$.

A point $P$ moves around the unit circle centered at the origin $O(0,0)$ with coordinates $P(\cos\theta,\sin\theta)$.

At a certain instant, the $x$-coordinate of $P$ is $-\dfrac{\sqrt{3}}{2}$. Find all possible values of $\sin\theta$.

If $P$ lies in the third quadrant, find the exact values of $\sin(2\theta)$, $\cos(2\theta)$, and $\tan(2\theta)$.

Verify that the identity $1+\tan^2(2\theta)=\sec^2(2\theta)$ holds for your values.

A quantity $Q$ measured at $P$ is proportional to $(\cos\theta-\sin\theta)^2$. Determine the maximum and minimum possible values of $Q$, giving exact values.

Hence find the range of $\theta$ (in radians, $0\le\theta<2\pi$) for which the quantity $Q$ is greater than $1$.

From identity $\sin^2\theta+\cos^2\theta=1\Rightarrow \cos^2\theta=1-(-\tfrac{4}{5})^2=\tfrac{9}{25}$. M1 $\cos\theta=\pm\dfrac{3}{5}$. A1

If $P$ lies in the fourth quadrant, find the exact values of $\sin(2\theta)$, $\cos(2\theta)$, and $\tan(2\theta)$.

Verify that the identity $\cot^2(2\theta)+1=\csc^2(2\theta)$ holds for your values.

A rotating light beam at $P$ has intensity $I$ proportional to $(\sqrt{5}\sin\theta+\sqrt{5}\cos\theta)^2$. Determine the maximum and minimum possible intensities, giving exact values.

Hence find the range of $\theta$ (in radians, $0\le\theta<2\pi$) for which the intensity is less than 5.

Find all possible values of $\sin\theta$.

If $P$ lies in the first quadrant, find the exact values of $\sin(2\theta)$, $\cos(2\theta)$, and $\tan(2\theta)$.

Verify that the identity $1+\cot^2(2\theta)=\csc^2(2\theta)$ holds for your values.

A quantity $Q$ measured at $P$ is proportional to $(\sin\theta+\cos\theta)^2$. Determine the maximum and minimum possible values of $Q$, giving exact values.

Hence find the range of $\theta$ (in radians, $0\le\theta<2\pi$) for which the quantity $Q$ is greater than $0$.

Find all possible values of $\sin\theta$.

If $P$ lies in the second quadrant, find the exact values of $\sin(2\theta)$, $\cos(2\theta)$, and $\tan(2\theta)$.

Verify that the identity $1+\tan^2(2\theta)=\sec^2(2\theta)$ holds for your values.

A rotating beacon at the point $P$ emits a light beam of intensity proportional to $(\sin\theta+\cos\theta)^2$.
As $\theta$ increases from $0$ to $2\pi$, determine the maximum and minimum possible intensities, giving exact values.

Hence find the range of $\theta$ (in radians, $0\le\theta<2\pi$) for which the intensity exceeds 1.5.