3.2 Trigonometry in triangles questions for IB Maths AA

Find the length of the diagonal $[\mathrm{A}\mathrm{G}]$.

Let $\theta$ be the angle between $[\mathrm{A}\mathrm{G}]$ and the base $\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$. Find the exact value of $\cos \theta$.

Find the exact value of $\cos \mathrm{G}\widehat{\mathrm{A}}\mathrm{B}$.

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

Find the angle between $[\mathrm{B}\mathrm{E}]$ and the base of the ramp.

In the first case, the ship's position vector is $\mathbf{r}=\left(\begin{array}{c}8\\ 0\end{array}\right)+t\left(\begin{array}{c}0\\ 6\end{array}\right)$, and the patrol boat travels at $10{\text{ km h}}^{-1}$.

Write down the speed of the ship.

Show that the patrol boat meets the ship when $t=1$.

Find the velocity vector of the patrol boat and the bearing on which it travels.

In general, let the ship's position vector be $\mathbf{r}=\mathbf{a}+t\mathbf{u}$, where $\mathrm{\mid }\mathbf{a}\mathrm{\mid }=d>0$ and $\mathrm{\mid }\mathbf{u}\mathrm{\mid }=w>0$. Let $\alpha$ be the angle between $\mathbf{a}$ and $\mathbf{u}$, where $0\le \alpha \le \pi$.

Show that the patrol boat can meet the ship at time $t>0$ if and only if

Suppose that $v>w$. Show that the patrol boat can always intercept the ship, and that there is exactly one possible time of interception.

Suppose now that $v<w$. Show that interception is only possible if $\cos \alpha <0$.

Show that interception is also only possible if $v\ge w\sin \alpha$.

Consider the case $v=w$. Find an expression for the time of interception in terms of $d$, $w$ and $\alpha$, and state the condition on $\alpha$ for interception to be possible.

In the second case, the ship's position vector is $\mathbf{r}=\left(\begin{array}{c}20\\ 15\end{array}\right)+t\left(\begin{array}{c}-10\\ 0\end{array}\right)$.

Find the minimum speed at which the patrol boat can intercept the ship.

The patrol boat travels at $8{\text{ km h}}^{-1}$. Find the earliest time at which it can intercept the ship.

The patrol boat travels at the minimum speed found in part (f)(i).

Find the time and the point of interception, and show that the velocity of the patrol boat is perpendicular to the initial position vector of the ship.

Show that triangle $\mathrm{A}\mathrm{B}\mathrm{C}$ is isosceles.

Write down the coordinates of $\mathrm{M}$, the midpoint of $[\mathrm{B}\mathrm{C}]$.

Hence find the exact area of triangle $\mathrm{A}\mathrm{B}\mathrm{C}$.

A ladder of length $5$ m slides so that its foot moves away from the wall at a constant speed of $0.3$ m s${}^{-1}$.

Find the speed at which the top of the ladder moves down the wall when the foot is $3$ m from the wall.

Explain why this model is unrealistic as the top of the ladder approaches the ground.

Consider a ladder of length $L$ m. Let M be the midpoint of [AB].

Show that, for every position of the ladder, M lies on a circle, and state the centre and radius of this circle.

Let $\theta$ be the angle between the ladder and the ground, where $0<\theta <{\displaystyle \frac{\pi }{2}}$.

Show that the line (AB) has equation $x\sin \theta +y\cos \theta =L\sin \theta \cos \theta$.

Consider the curve $E$ with equation $x^{\frac{2}{3}}+y^{\frac{2}{3}}=L^{\frac{2}{3}}$, for $x>0$, $y>0$.

Use implicit differentiation to show that ${\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x}}=-{\left({\displaystyle \frac{y}{x}}\right)}^{\frac{1}{3}}$.

The point P has coordinates $(L{\cos }^3\theta ,L{\sin }^3\theta )$, where $0<\theta <{\displaystyle \frac{\pi }{2}}$.

Show that P lies on $E$.

Show that the tangent to $E$ at P is the line (AB) from part (c).

Explain briefly the significance of the result in part (f) for the sliding ladder.

Now consider the family of curves $x^n+y^n=1$, for $x>0$, $y>0$, where $n>0$. The tangent to such a curve at the point $(p,q)$ meets the $x$-axis at X and the $y$-axis at Y.

Show that the tangent at $(p,q)$ has equation $p^{n-1}x+q^{n-1}y=1$.

Hence show that ${\mathrm{X}\mathrm{Y}}^2=p^{2-2n}+q^{2-2n}$.

Show that when $n={\displaystyle \frac{2}{3}}$, the length XY is the same for every point $(p,q)$ on the curve.

By considering the points $({\displaystyle \frac{1}{\sqrt{2}}},{\displaystyle \frac{1}{\sqrt{2}}})$ and $({\displaystyle \frac{\sqrt{3}}{2}},{\displaystyle \frac{1}{2}})$, show that the length XY is not constant when $n=2$.