3.6 Trigonometric identities questions for IB Maths AA

Show that the equation can be written as $3{\sin }^{2}x+2\sin x-1=0$.

Hence, solve the equation $3\cos 2x=4\sin x+1$, for $0\le x\le 2\pi$.

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 area of triangle $\mathrm{A}\mathrm{B}\mathrm{C}$.

Find the exact length of $[\mathrm{B}\mathrm{C}]$.

Express $f(x)$ in the form $a\sin (bx)$, where $a,b\in {\mathbb{Z}}^{+}$.

Write down the period of $f$.

Write down the range of $f$.

Solve the equation $f(x)=1.8$.

Find the value of $I_0$, the value of $I_1$ and the value of $I_2$.

By writing ${\cos }^{n}x={\cos }^{n-1}x\cos x$ and using integration by parts, show that $I_n={\displaystyle \frac{n-1}{n}}{I}_{n-2}$, for $n\ge 2$.

Find the value of $I_4$ and the value of $I_5$.

Prove by mathematical induction that $I_{2n}={\displaystyle \frac{(2n)!\pi }{2^{2n+1}(n!{)}^2}}$, for $n\in \mathbb{N}$.

Prove by mathematical induction that $n{I}_n{I}_{n-1}={\displaystyle \frac{\pi }{2}}$, for $n\in {\mathbb{Z}}^{+}$.

Explain why $I_n\le I_{n-1}$, for $n\in {\mathbb{Z}}^{+}$.

Hence show that $\sqrt{{\displaystyle \frac{\pi }{2(n+1)}}}\le I_n\le \sqrt{{\displaystyle \frac{\pi }{2n}}}$, for $n\in {\mathbb{Z}}^{+}$.

Use your GDC to find the value of $I_{20}$, and verify that it satisfies the inequalities in part (g).