3.9 Reciprocal and inverse trigonometric functions questions for IB Maths AA

Find the exact value of $\arctan 2+\arctan 3$.

Solve the equation $\arctan x+\arctan 2x={\displaystyle \frac{\pi }{4}}$.

By considering $\tan (\beta -\alpha )$, find the acute angle between $L_1$ and $L_2$. Give your answer in degrees.

The line $L_3$ passes through the point $(0,2)$ and makes an acute angle of $40\mathrm{°}$ with $L_1$.

Find the possible values of the gradient of $L_3$.

Find the exact value of $f\left(\frac{\pi }{6}\right)$.

Find the exact value of $f\left(\frac{5\pi }{6}\right)$.

Find the exact value of $f\left(\frac{7\pi }{6}\right)$.

Use your GDC to sketch the graph of $y=f(x)$ for $-\pi \le x\le 2\pi$. Clearly indicate the coordinates of any maximum and minimum points.

Show that $f(x)=\pi -x$ for $\frac{\pi }{2}\le x\le \frac{3\pi }{2}$.

Write down the period of $f$.

Hence write down an expression for $f(x)$ for $\frac{3\pi }{2}\le x\le \frac{5\pi }{2}$.

Show that $f$ is an odd function.

Write down an expression for $g(x)$ for $0\le x\le \pi$.

Show that $g(x)=2\pi -x$ for $\pi \le x\le 2\pi$.

Let $h(x)=f(x)+g(x)$, for $0\le x\le 2\pi$.

Find an expression for $h(x)$ on each of the intervals $0\le x\le \frac{\pi }{2}$, $\frac{\pi }{2}\le x\le \pi$, $\pi \le x\le \frac{3\pi }{2}$ and $\frac{3\pi }{2}\le x\le 2\pi$.

Find all the solutions of the equation $f(x)={\displaystyle \frac{x}{2}}$, for $x\ge 0$.

Show that, for $\cos x\mathrm{\ne }0$, $f^{\mathrm{\prime }}(x)={\displaystyle \frac{\cos x}{\mathrm{\mid }\cos x\mathrm{\mid }}}$.

Hence explain why $f$ is not differentiable at $x=\frac{\pi }{2}$.

Find the exact value of ${\displaystyle {\int }_0^{10\pi }\mathrm{\mid }f(x)\mathrm{\mid }dx}$.

Show that $\arctan p-\arctan q=\arctan \left({\displaystyle \frac{p-q}{1+pq}}\right)$, for $p,q\ge 0$.

Hence show that $\arctan (n+2)-\arctan n=\arctan \left({\displaystyle \frac{2}{(n+1{)}^2}}\right)$, for $n\ge 0$.

Show that $S_2={\displaystyle \frac{\pi }{2}}$.

Prove by mathematical induction that $S_n=\arctan n+\arctan (n+1)-{\displaystyle \frac{\pi }{4}}$, for $n\in {\mathbb{Z}}^{+}$.

Find ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }{S}_n}$.

Write down the equations of the vertical asymptotes of the graph of $y=f(x)$.

Find the range of $f$ for $\frac{\pi }{4}<x<\frac{3\pi }{4}$.

Solve the equation $f(x)=x$.

By differentiating $\tan y=x$ implicitly with respect to $x$, show that ${\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}}(\arctan x)={\displaystyle \frac{1}{1+x^2}}$.

Let $g(x)=\arctan x+\arctan \left({\displaystyle \frac{1}{x}}\right)$, for $x\mathrm{\ne }0$.

Show that $g^{\mathrm{\prime }}(x)=0$.

Hence find the value of $g(x)$ for $x>0$ and the value of $g(x)$ for $x<0$.

Explain why $g^{\mathrm{\prime }}(x)=0$ for all $x\mathrm{\ne }0$, but $g$ is not a constant function.

Let $a$ be a positive constant and let $h(x)=\arctan \left({\displaystyle \frac{a+x}{1-ax}}\right)-\arctan x$, for $x<{\displaystyle \frac{1}{a}}$.

Show that $h^{\mathrm{\prime }}(x)=0$.

Hence show that $\arctan a+\arctan x=\arctan \left({\displaystyle \frac{a+x}{1-ax}}\right)$ for all $x<{\displaystyle \frac{1}{a}}$.

Use part (c) to show that $\arctan \left({\displaystyle \frac{1}{2}}\right)+\arctan \left({\displaystyle \frac{1}{3}}\right)={\displaystyle \frac{\pi }{4}}$.

Use part (c) to show that, for integers $n\ge 2$,

Let $S_N={\displaystyle \sum _{n=2}^N\arctan \left(\frac{2}{n^2}\right)}$, where $N\ge 2$.

Hence show that $S_N={\displaystyle \frac{\pi }{4}}+\arctan \left({\displaystyle \frac{1}{2}}\right)-\arctan \left({\displaystyle \frac{1}{N}}\right)-\arctan \left({\displaystyle \frac{1}{N+1}}\right)$.

Hence show that ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\arctan \left(\frac{2}{n^2}\right)=\frac{3\pi }{4}}$.