IB Mathematics Analysis and Approaches Topic AHL…

In the diagram below, AD is perpendicular to $\mathrm{BC}, \mathrm{CD}=4, \mathrm{BD}=2$ and $\mathrm{AD}=3$ . Also $∠ C$ and .

(a) Find the exact value of $\cos (\alpha+\beta)$ .

(b) Find the exact value of $\sin (\alpha+\beta)$ .

(e) Hence find all possible values of $\theta$ , in radians with $0\le\theta<2\pi$ , for which $Q>1$ .

A digital display, $BC$ , with height two metres is mounted in a wall. The lower end of the display at $B$ is one metre above the eye level of an observer. The angles of elevation from the observer's eye at $O$ to the top and bottom of the display are $\alpha$ and $\beta$ radians respectively. The horizontal distance from the observer's eye to the wall containing the display is $x$ metres. The observer's viewing angle, $\angle BOC$ , is $\theta$ radians, as illustrated below. diagram of observer viewing a wall-mounted display

(a) Show that $\theta=\arctan \frac{3}{x}-\arctan \frac{1}{x}$

(b) Express $\theta$ as $\theta=\arctan \frac{a x}{x^{2}+b}$ , where $a, b \in R$

(d) Find where the observer should stand so that the viewing angle is $25^{\circ}$ .

A point $P$ moves around the unit circle centered at the origin $O(0,0)$ with coordinates $P(\cos\theta,\sin\theta)$ .

The $x$ -coordinate of $P$ is $-\dfrac{\sqrt{3}}{2}$ .

(a) Find all possible values of $\sin\theta$ .

(b) If $P$ lies in the third quadrant, find the exact values of $\sin(2\theta)$ , $\cos(2\theta)$ , and $\tan(2\theta)$ .

(a) Use the compound angle formula $\sin(A + B) = \sin A\cos B + \cos A\sin B$ to find the exact value of $\sin 75^\circ$ .

(b) Find the exact values of:

(i) $\cos 75^\circ$ ;

(ii) $\tan 75^\circ$ .

(d) Solve $2\sin(x + 60^\circ) = \sqrt{3}$ for $0 \leq x < 360^\circ$ .

(a) Using the formula for $\cos(A+B)$ and suitable double angle identities, prove the following identity: $\cos 4\theta=1-8\sin^2\theta\cos^2\theta$ .

(b) Find an expression for $\cos 6\theta$ in terms of $\sin\theta$ and $\cos\theta$ only.

(d) Let $Q=(\cos\theta-\sin\theta)^2$ . the maximum and minimum possible values of $Q$ , giving exact values.

IB Mathematics Analysis and Approaches AHL 3.10—compound Angle Identities Questionbank

IB Mathematics Analysis and Approaches AHL 3.10—compound Angle Identities Questionbank

IB Mathematics Analysis and Approaches AHL 3.10—compound Angle Identities Questionbank

IB Mathematics Analysis and Approaches AHL 3.10—compound Angle Identities Questionbank

IB Mathematics Analysis and Approaches (AA) Topic AHL 3.10—compound Angle Identities Questionbank