Geometry and trigonometry: 87 IB Maths AA HL practice questions

It is given that $\csc \theta = \frac{3}{2}$, where $\frac{\pi}{2} < \theta < \frac{3\pi}{2}$. Find the exact value of $\cot \theta$.[4]

The points $A$ and $B$ have position vectors $\overrightarrow{OA} = \begin{pmatrix} 1 \\ 2 \\ -2 \end{pmatrix}$ and $\overrightarrow{OB} = \begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix}$.

(a) Find $\overrightarrow{OA} \times \overrightarrow{OB}$.[2]

(b) Hence find the area of the triangle $OAB$.[2]

Consider the points $A(1, 2, 3)$, $B(1, 0, 5)$ and $C(2, -1, 4)$.

(a) Find $\overrightarrow{AB} \times \overrightarrow{AC}$.[4]

(b) Hence find the area of the triangle $ABC$.[2]

Solve the equation $2\cos^2 x + 5\sin x = 4$, $0 \leq x \leq 2\pi$.[7]

The following diagram shows a circle with centre $O$.

Points $A$, $B$ and $C$ lie on the circle.

The area of triangle $AOB$ is $26 \text{ cm}^2$ and $\angle AOB = 2.51$ radians.

Find the length of arc $ACB$.[5]

Two boats, $A$ and $B$, move so that at time $t$ hours, their position vectors, in kilometres, are $\mathbf{r}_a = (9t)\mathbf{i} + (3 - 6t)\mathbf{j}$ and $\mathbf{r}_b = (7 - 4t)\mathbf{i} + (7t - 6)\mathbf{j}$.

(a) Find the coordinates of the common point of the paths of the two boats.[4]

(b) Show that the boats do not collide.[2]

Consider the two planes

$$\pi_1: 4x + 2y - z = 8$$

$$\pi_2: x + 3y + 3z = 3$$

Find the angle between $\pi_1$ and $\pi_2$, giving your answer correct to the nearest degree.[6]

Solve $\cos 2x = \sin x$, where $-\pi \le x \le \pi$.[6]

Find the values of $x$ for which the vectors $\begin{pmatrix} 1 \\ 2\cos x \\ 0 \end{pmatrix}$ and $\begin{pmatrix} -1 \\ 2\sin x \\ 1 \end{pmatrix}$ are perpendicular, $0 \le x \le \frac{\pi}{2}$[5]

In triangle $ABC$, $AB = 5$, $BC = 14$ and $AC = 11$. Find all the interior angles of the triangle. Give your answers in degrees to one decimal place.[5]

Given that $\frac{\pi}{2} < \alpha < \pi$ and $\cos\alpha = -\frac{3}{4}$, find the value of $\sin 2\alpha$.[4]

Triangle ABC has $AB = 5$ cm, $BC = 6$ cm and area $10$ cm².

(a) Find $\sin B$.[2]

(b) Hence, find the two possible values of AC, giving your answers correct to two decimal places.[4]

The logo, for a company that makes chocolate, is a sector of a circle of radius 2 cm, shown as shaded in the diagram. The area of the logo is $3\pi$ cm².

a. Find, in radians, the value of the angle $\theta$, as indicated on the diagram.[3]

b. Find the total length of the perimeter of the logo.[2]

The acute angle between the vectors $3i - 4j - 5k$ and $5i - 4j + 3k$ is denoted by $\theta$. Find $\cos\theta$.[4]

ABCD is a quadrilateral where $AB = 6.5$, $BC = 9.1$, $CD = 10.4$, $DA = 7.8$ and $\angle CDA = 90°$. Find $\angle ABC$, giving your answer correct to the nearest degree.[5]