Exercises: Product-to-sum, sum-to-product and 3D trigonometry

Use the formula $2\sin A\cos B = \sin(A+B) + \sin(A-B)$ to express $2\sin 5\theta\cos 3\theta$ as a sum of sines.

Use the formula $\cos P + \cos Q = 2\cos\frac{P+Q}{2}\cos\frac{P-Q}{2}$ to write $\cos 7x + \cos x$ as a product.

A vertical pole $AB$ of height 6 m stands on level ground at $B$. A point $C$ on the ground is 8 m from $B$. Find the length $AC$ and the angle of elevation of $A$ from $C$, to the nearest $0.1^\circ$.

1. a) Express $\cos 4\theta\cos 2\theta$ as a sum of cosines.
2. b) Hence find the exact value of $\cos 75^\circ\cos 15^\circ$.

1. a) Express $\sin 75^\circ + \sin 15^\circ$ as a product and hence find its exact value.
2. b) Solve $\sin 3x + \sin x = 0$ for $0^\circ \le x \le 180^\circ$.

A radio mast $TB$ of height 40 m stands on level ground at $B$. From a point $A$ due west of $B$ the angle of elevation of $T$ is $28^\circ$. From a point $C$ due south of $B$ the angle of elevation of $T$ is $35^\circ$.

1. a) Find $AB$ and $CB$ to the nearest 0.1 m.
2. b) Find the distance $AC$.
3. c) Find the bearing of $C$ from $A$, to the nearest degree.

1. a) Prove that $\sin A + \sin B = 2\sin\frac{A+B}{2}\cos\frac{A-B}{2}$ using the formulae for $\sin(X\pm Y)$.
2. b) Hence prove that $\dfrac{\sin 5\theta + \sin\theta}{\cos 5\theta + \cos\theta} = \tan 3\theta$.
3. c) Solve $\sin 5\theta + \sin\theta = \cos 5\theta + \cos\theta$ for $0^\circ\le\theta\le 90^\circ$.

1. a) Show that $\cos 3\theta + \cos\theta = 2\cos2\theta\cos\theta$.
2. b) Hence solve $\cos 3\theta + \cos \theta + \cos 2\theta = 0$ for $0^\circ \le\theta\le 360^\circ$.

A tetrahedral frame $ABCD$ has base $ABC$ in a horizontal plane, with $AB = 10$ m, $BC = 8$ m and $\angle ABC = 90^\circ$. The vertex $D$ is vertically above $B$ with $BD = 6$ m.

1. a) Find the length $AC$, $AD$ and $CD$.
2. b) Find the angle between the face $ACD$ and the base $ABC$, to the nearest $0.1^\circ$.
3. c) Find the angle between the edge $AD$ and the base, and the area of triangle $ACD$.