Compound and double angles Worked Solutions — A-Level Maths

Fully worked, step-by-step solutions to A-Level Compound and double angles questions. See exactly how to solve problems on addition formulae, exact values, double angle, sin 2x.

addition formulaeexact valuesdouble anglesin 2xcos 2xcompound angles
A-Level70 questionsStep-by-step solutions
Question 1
2 markseasy
Using an addition formula, find the exact value of sin⁡75∘\sin 75^{\circ}.

Worked solution

  1. Write the angle as a sum or difference of two special angles

    75∘=45∘+30∘75^{\circ}=45^{\circ}+30^{\circ}

    Choose familiar angles (30, 45, 60 degrees) whose exact ratios are known.

  2. State the relevant addition formula

    sin⁡(A+B)≡sin⁡Acos⁡B+cos⁡Asin⁡B\sin(A+B)\equiv \sin A\cos B+\cos A\sin B

    The compound-angle formula rewrites the ratio of a sum in terms of the parts.

  3. State the exact value

    sin⁡75∘=24+64\sin 75^{\circ}=\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}

    This is the required exact value.

Answer
24+64\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}
Question 2
2 markseasy
Using an addition formula, find the exact value of cos⁡15∘\cos 15^{\circ}.

Worked solution

  1. Write the angle as a sum or difference of two special angles

    15∘=45∘−30∘15^{\circ}=45^{\circ}-30^{\circ}

    Choose familiar angles (30, 45, 60 degrees) whose exact ratios are known.

  2. State the relevant addition formula

    cos⁡(A−B)≡cos⁡Acos⁡B+sin⁡Asin⁡B\cos(A-B)\equiv \cos A\cos B+\sin A\sin B

    The compound-angle formula rewrites the ratio of a sum in terms of the parts.

  3. State the exact value

    cos⁡15∘=24+64\cos 15^{\circ}=\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}

    This is the required exact value.

Answer
24+64\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}
Question 3
2 markseasy
Using an addition formula, find the exact value of sin⁡15∘\sin 15^{\circ}.

Worked solution

  1. Write the angle as a sum or difference of two special angles

    15∘=45∘−30∘15^{\circ}=45^{\circ}-30^{\circ}

    Choose familiar angles (30, 45, 60 degrees) whose exact ratios are known.

  2. State the relevant addition formula

    sin⁡(A−B)≡sin⁡Acos⁡B−cos⁡Asin⁡B\sin(A-B)\equiv \sin A\cos B-\cos A\sin B

    The compound-angle formula rewrites the ratio of a sum in terms of the parts.

  3. State the exact value

    sin⁡15∘=−24+64\sin 15^{\circ}=- \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}

    This is the required exact value.

Answer
−24+64- \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}
Question 4
2 markseasy
Using an addition formula, find the exact value of cos⁡75∘\cos 75^{\circ}.

Worked solution

  1. Write the angle as a sum or difference of two special angles

    75∘=45∘+30∘75^{\circ}=45^{\circ}+30^{\circ}

    Choose familiar angles (30, 45, 60 degrees) whose exact ratios are known.

  2. State the relevant addition formula

    cos⁡(A+B)≡cos⁡Acos⁡B−sin⁡Asin⁡B\cos(A+B)\equiv \cos A\cos B-\sin A\sin B

    The compound-angle formula rewrites the ratio of a sum in terms of the parts.

  3. State the exact value

    cos⁡75∘=−24+64\cos 75^{\circ}=- \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}

    This is the required exact value.

Answer
−24+64- \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}
Question 5
2 markseasy
Using an addition formula, find the exact value of sin⁡105∘\sin 105^{\circ}.

Worked solution

  1. Write the angle as a sum or difference of two special angles

    105∘=60∘+45∘105^{\circ}=60^{\circ}+45^{\circ}

    Choose familiar angles (30, 45, 60 degrees) whose exact ratios are known.

  2. State the relevant addition formula

    sin⁡(A+B)≡sin⁡Acos⁡B+cos⁡Asin⁡B\sin(A+B)\equiv \sin A\cos B+\cos A\sin B

    The compound-angle formula rewrites the ratio of a sum in terms of the parts.

  3. State the exact value

    sin⁡105∘=24+64\sin 105^{\circ}=\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}

    This is the required exact value.

Answer
24+64\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}

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