SL 3.6—Pythagorean identity, double angles Questionbank

Using the binomial expansion, find the 2nd and 4th terms ($T_2,T_4$ respectively) in ascending powers of $\sin x$

Hence show that $T_2-T_4=\sin(4x)$

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 $\frac{a+b\sqrt{c}}{d}$

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 $0$.

Show that $\sin 15^{\circ}=\frac{\sqrt{2-\sqrt{3}}}{2}$

Find a similar expression for $\cos 15^{\circ}$

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.

Show that $f(x)=\sin x$

Let $\sin x=\frac{2}{3}$. Find the exact value of $f(2 x)$

At the point of observation, the $\sin\theta$ component is measured as $-\dfrac{4}{5}$. Find all possible values of the $\cos\theta$ component.

Find values of $\sin2\theta$, $\cos2\theta$, $\tan2\theta$

Confirm that the identity $1+\cot^2(2\theta)=\csc^2(2\theta)$ holds true for the calculated double angle values.

The observed luminosity $L$ is proportional to $(\sqrt{5}\sin\theta+\sqrt{5}\cos\theta)^2$. Over a full rotation ($0\le\theta<2\pi$), determine the maximum and minimum possible luminosities, giving exact values.

Find the angular span (range of $\theta$ in radians, $0\le\theta<2\pi$) for which the luminosity $L$ is less than $5$.