3.10 Compound angle identities questions for IB Maths AA

By writing $\frac{7\pi }{12}$ as $\frac{\pi }{3}+\frac{\pi }{4}$, find the exact value of $\cos \frac{7\pi }{12}$.

Find the exact value of $\tan \frac{7\pi }{12}$, giving your answer in the form $a+b\sqrt{3}$, where $a,b\in \mathbb{Z}$.

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 $L_3$.

Use a double angle identity to show that $T_2(x)=2x^2-1$.

By expanding $\cos ((n+1)\theta )$ and $\cos ((n-1)\theta )$, show that $\cos ((n+1)\theta )+\cos ((n-1)\theta )\equiv 2\cos \theta \cos n\theta$.

Hence show that $T_{n+1}(x)=2x{T}_n(x)-T_{n-1}(x)$, for $n\in {\mathbb{Z}}^{+}$.

Hence find $T_3(x)$ and $T_4(x)$.

Use your GDC to sketch the graph of $y=T_4(x)$ for $-1\le x\le 1$, and write down the range of $T_4$ on this interval.

Write down the leading coefficient of $T_n(x)$ for $n=1,2,3,4$, and hence suggest an expression for the leading coefficient of $T_n(x)$, for $n\in {\mathbb{Z}}^{+}$.

Prove by mathematical induction that, for $n\in {\mathbb{Z}}^{+}$, $T_n(x)$ is a polynomial of degree $n$ with leading coefficient $2^{n-1}$.

Show that the $n$ roots of $T_n(x)=0$ are $x=\cos \left({\displaystyle \frac{(2k-1)\pi }{2n}}\right)$, for $k=1,2,\dots ,n$.

Hence find the exact value of $\cos {\displaystyle \frac{\pi }{8}}$.

By considering the product of the roots of $T_4(x)=0$, find the exact value of $\cos {\displaystyle \frac{\pi }{8}}\cos {\displaystyle \frac{3\pi }{8}}$.

Find the exact value of $\sin (A+B)$.

Find the exact value of $\tan (A-B)$.

Write down the maximum height of cabin P and the first time at which it reaches this height.

Let $d(t)=h_Q(t)-h_P(t)$. Show that $d(t)=10\cos \left({\displaystyle \frac{\pi t}{6}}\right)+10\sqrt{3}\sin \left({\displaystyle \frac{\pi t}{6}}\right)$.

By expanding $20\sin ({\displaystyle \frac{\pi t}{6}}+{\displaystyle \frac{\pi }{6}})$, show that $d(t)=20\sin ({\displaystyle \frac{\pi t}{6}}+{\displaystyle \frac{\pi }{6}})$.

Write down the greatest value of $d(t)$, and find the first time at which it occurs.

Find the length of time, during the first revolution, for which cabin Q is more than 15 m higher than cabin P.

Find the rate of change of $d$ when $t=5$.

Use your graphic display calculator to find the value of $I_1$, the value of $I_2$ and the value of $I_3$.

Suggest an exact value for $I_n$.

Show that $\sin ({\displaystyle \frac{\pi }{2}}-x)=\cos x$ and $\cos ({\displaystyle \frac{\pi }{2}}-x)=\sin x$.

By using the substitution $u={\displaystyle \frac{\pi }{2}}-x$, show that

Hence prove that your suggestion in part (b) is correct for all $n\in {\mathbb{Z}}^{+}$.

Now consider a function $f$ that is continuous for $0\le x\le a$, where $a>0$, and such that $f(x)+f(a-x)\mathrm{\ne }0$ for $0\le x\le a$.

Show that

Hence find the value of ${\displaystyle {\int }_0^4\frac{\sqrt{x}}{\sqrt{x}+\sqrt{4-x}}dx}$.

Now let $g$ be an even function that is continuous for $-a\le x\le a$.

Show that

Hence find the exact value of ${\displaystyle {\int }_{-\frac{\pi }{3}}^{\frac{\pi }{3}}\frac{\cos x}{1+e^x}dx}$.