3.11 Symmetry of trigonometric graphs questions for IB Maths AA

Write down, in terms of $p$, the value of $\sin {\displaystyle \frac{8\pi }{9}}$.

Write down, in terms of $p$, the value of $\sin {\displaystyle \frac{10\pi }{9}}$.

Find, in terms of $p$, the value of $\cos {\displaystyle \frac{11\pi }{18}}$.

Find, in terms of $p$, the value of $\tan {\displaystyle \frac{17\pi }{9}}$.

Find the value of $\alpha$.

Write down, in terms of $\alpha$, the four solutions of $\cos x=0.35$ for $-2\pi \le x\le 2\pi$.

Hence solve the equation $\cos x=-0.35$ for $0\le x\le 2\pi$.

Hence solve the equation $\cos 2x=-0.35$ for $0\le x\le \pi$.

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 ${\displaystyle {\int }_{-\frac{\pi }{3}}^{\frac{\pi }{3}}\frac{\cos x}{1+e^x}dx}$.

Show that $g$ is an odd function.

Hence find the exact value of ${\displaystyle {\int }_{-\frac{\pi }{4}}^{\frac{\pi }{4}}(x^2\sin x+\cos 2x)dx}$.

Write down the time taken for the wheel to complete one rotation.

The model can also be written in the form $h(t)=18\sin ({\displaystyle \frac{\pi }{15}}(t-q))+20$, where $q>0$. Find the smallest possible value of $q$.

Find the first time at which the capsule is 30 metres above the ground.

By using the symmetry of the graph of $h$, find the length of time, during each rotation, for which the capsule is more than 30 metres above the ground.

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 ${\displaystyle {\int }_0^{10\pi }\mathrm{\mid }f(x)\mathrm{\mid }dx}$.