IB Maths AA HL Ahl 3 10 Hl Compound Angle Identities Questions

Practise using compound- and double-angle identities to simplify, prove and solve trigonometric expressions while respecting signs, quadrants and denominators.

Exam points

Use compound-angle identities for sine, cosine and tangent to expand or simplify expressions and exact values, choosing suitable known angles.
Apply double-angle identities, including tan 2θ, to derive or evaluate trigonometric expressions and reduce equations to an algebraic form.
Prove trigonometric identities by expanding both sides or converting between sine/cosine and tangent forms, while respecting denominator restrictions.
Solve or transform compound-angle equations and models, selecting admissible signs, quadrants and parameter values.

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Study AHL 3.10 (HL)—Compound-angle identities conceptsOpen the corresponding Concept lesson for explanations, examples and exam connections.

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Show that $\sin 105^{\circ}+\cos 105^{\circ}=\frac{1}{\sqrt{2}}$ .

(b) EITHER
choosing two appropriate angles, for example $60^{\circ}$ and $45^{\circ}$ $\sin 105^{\circ}=\sin 60^{\circ} \cos 45^{\circ}+\cos 60^{\circ} \sin 45^{\circ}$ and
$\cos 105^{\circ}=\cos 60^{\circ} \cos 45^{\circ}-\sin 60^{\circ} \sin 45^{\circ}$ $\sin 105^{\circ}+\cos 105^{\circ}=\frac{\sqrt{3}}{2} \times \frac{1}{\sqrt{2}}+\frac{1}{2} \times \frac{1}{\sqrt{2}}+\frac{1}{2} \times \frac{1}{\sqrt{2}}-\frac{\sqrt{3}}{2} \times \frac{1}{\sqrt{2}}$ $=\frac{1}{\sqrt{2}}$
OR
attempt to square the expression
$\left(\sin 105^{\circ}+\cos 105^{\circ}\right)^{2}=\sin ^{2} 105^{\circ}+2 \sin 105^{\circ} \cos 105^{\circ}+\cos ^{2} 105^{\circ}$ $\left(\sin 105^{\circ}+\cos 105^{\circ}\right)^{2}=1+\sin 210^{\circ}$ $=\frac{1}{2}$ $\sin 105^{\circ}+\cos 105^{\circ}=\frac{1}{\sqrt{2}}$

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IB Maths AA HL Ahl 3 10 Hl Compound Angle Identities Questions question 1