AHL 3.9—Reciprocal trig ratios and their pythagorean identities. Inverse circular functions Questionbank

$\tan x+\cot x=2 \csc 2 x$

Find the coordinates of the local maximum and local minimum points on the graph of $y=\tan 2 x+\cot 2 x$, for $0 \leq x \leq \frac{\pi}{2}$

Find the solution of the equation $\csc 2 x=1.5 \tan x-0.5$, for $0 \leq x \leq \frac{\pi}{2}$

Sketch the graph of $y=f(x)$, in the interval $\frac{-\pi}{2} \leq x \leq \frac{\pi}{2}$ and state $x$ -intercepts, the equations of the asymptotes and the coordinates of the maximum and minimum points.

Show that roots of the equation $f(x)=0$ satisfy the equation $2 \cos ^{3} x-2 \cos ^{2} x-2 \cos x+1=0$

Show that $f(\pi-x)+f(\pi+x)=0$

Sketch the graph of $y=\cos (4 x)$, in the interval $0 \leq x \leq \pi$

On the same diagram sketch the graph of $y=\sec (4 x)$, in the interval $0 \leq x \leq \pi$, indicating clearly the equations of any asymptotes.

Use your sketch to solve :

$i$. the equation $\sec (4 x)=-1$ in the interval $0 \leq x \leq \pi$

$i i$. the inequality $\sec (4 x)<0$ in the interval $0 \leq x \leq \pi$

$\tan (\arccos (0.1))$

Find the value of $\sin (2 \arctan (5))$

Solve for $x$ in $16 \sec \left(\frac{\pi x}{36}\right)-32=-6$

Sketch the graph of $y=\sin (\pi x)$, in the interval $0 \leq x \leq 4$

On the same diagram sketch the graph of $y=f(x)$, in the interval $0 \leq x \leq 4$, indicating clearly the equations of any asymptotes.

Solve the equation $f(x)=1$, for $0 \leq x \leq 4$

Write down the number of solutions of the equation $f(x)=2$, for $0 \leq x \leq 4$

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

Solve, for $0\le x<2\pi$, the equation $\sec x+\tan x=2.4$.
(You may use $\sec x+\tan x=\dfrac{1+\sin x}{\cos x}=\tan\!\big(\tfrac{\pi}{4}+\tfrac{x}{2}\big)$.)
Give answers to 3 s.f.

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

A line through the origin $y=mx$ makes an angle $\alpha$ with the positive $x$-axis, so $m=\tan\alpha$.
Find all $\alpha\in[0,\pi)$ such that $\sec(2\alpha)=1.6$, and the corresponding slopes $m$ (3 s.f.).