AHL 3.11—Relationships between trig functions Questionbank

Show that the expression equal to $2\cos(x)\sin(y)$

Hence, using mathematical induction and the above identity, prove that $\sum_{r=1}^{n} \cos((2r-1)\alpha) = \frac{\sin(2n\alpha)}{2\sin\alpha}$ for $n \in \mathbb{Z}^+$.

Prove that $1 + \tan^2\theta = \sec^2\theta$ and hence show that $(\sec\theta - \tan\theta)(\sec\theta + \tan\theta) = 1.$

Solve, for $0\le x<2\pi$, the equation $\sec x+\tan x=1.7$.
(You may use $\displaystyle \sec x+\tan x=\frac{1+\sin x}{\cos x}=\tan\!\bigl(\tfrac{\pi}{4}+\tfrac{x}{2}\bigr)$.)
Give your answer to 3 significant figures.

Let $\phi=\arccos(0.45)$. Find exact decimal values of $\sin\phi$ and $\sin(2\phi)$.
State the quadrant of $\phi$ and verify that $\cos(\pi-\phi)=-\cos\phi$.

A line through the origin has equation $y=mx$ and makes an angle $\alpha$ with the positive $x$-axis, so that $m=\tan\alpha$.
Find all $\alpha\in[0,\pi)$ such that $\sec(2\alpha)=1.5$, and the corresponding slopes $m$ (to 3 significant figures).

Prove $1+\tan^2\theta=\sec^2\theta$ and hence show that $\frac{\sec\theta+\tan\theta}{\sec\theta-\tan\theta}=\frac{1+\sin\theta}{1-\sin\theta}.$

Solve, for $0\le x<2\pi$, the equation $\sec x+\tan x=2.0$. \nGive your answer to 3 significant figures.

Let $\phi=\arccos(-0.28)$. Find exact decimal values of $\sin\phi$ and $\sin(2\phi)$. \nState the quadrant of $\phi$ and verify that $\cos(\pi-\phi)=-\cos\phi$.

A line through the origin has slope $m=\tan\alpha$. \nFind all $\alpha\in[0,\pi)$ such that $\sec(2\alpha)=1.6$, and the corresponding slopes $m$ (3 s.f.).

$\cos 5 x=\cos 3 x$,

Explain why $\cos (\pi-x)=-\cos x$

Hence, solve the equation $\cos 5 x+\cos 3 x=0$, in the interval $[0, \pi][6]$

Prove that $1+\cot^2\theta=\csc^2\theta$ and hence show that $(\csc\theta+\cot\theta)(\csc\theta-\cot\theta)=1.$

Solve, for $0<x<2\pi$, the equation $\csc x+\cot x=2.3$.
(You may use $\displaystyle \csc x+\cot x=\frac{1+\cos x}{\sin x}=\cot\!\bigl(\tfrac{x}{2}\bigr)$.)
Give your answer to 3 significant figures.

Let $\phi=\arccos(0.40)$. Find exact decimal values of $\sin\phi$ and $\sin(2\phi)$.
State the quadrant of $\phi$ and verify that $\cos(\pi-\phi)=-\cos\phi$.

A line through the origin has equation $y=mx$ and makes an angle $\alpha$ with the positive $x$-axis, so that $m=\tan\alpha$.
Find all $\alpha\in[0,\pi)$ such that $\csc(2\alpha)=1.8$, and the corresponding slopes $m$ (to 3 significant figures).