Proof 1
\( \sin (\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \)
\( \require{AMSsymbols} \begin{align} \angle RPN &= 90^{\circ}-\angle PNR \\ &= \angle RNO \\ &= \angle RNO \\ &= \angle NOQ \\ &= \alpha \\ \sin (\alpha + \beta) &= \sin \angle AOC \\ &= \displaystyle \frac{MP}{OP} \\ &= \frac{MR + RP}{OP} \\ &= \frac{QN}{OP} + \frac{RP}{OP} \\ &= \frac{QN}{\color{red}{ON}} \times \frac{\color{red}{ON}}{OP} + \frac{RP}{\color{red}{NP}} \times \frac{\color{red}{NP}}{OP} \\ &= \sin \alpha \cos \beta + \cos \angle RPN \sin \beta \\ \therefore \sin (\alpha + \beta) &= \sin \alpha \cos \beta + \cos \alpha \sin \beta \end{align} \)
Proof 2
\( \cos (\alpha + \beta) = \cos \alpha \cos \beta-\sin \alpha \sin \beta \)
\( \require{AMSsymbols} \begin{align} \cos (\alpha + \beta) &= \cos \angle AOC \\ &= \displaystyle \frac{OM}{OP} \\ &= \frac{OQ-MQ}{OP} \\ &= \frac{OQ}{OP} – \frac{RN}{OP} \\ &= \frac{OQ}{\color{red}{ON}} \times \frac{\color{red}{ON}}{OP} – \frac{RN}{\color{red}{NP}} \times \frac{\color{red}{NP}}{OP} \\ &= \cos \alpha \cos \beta-\sin \angle RPN \sin \beta \\ \therefore \cos (\alpha + \beta) &= \cos \alpha \cos \beta-\sin \alpha \sin \beta \end{align} \)
Proof 3
\( \tan (\alpha + \beta) = \displaystyle \frac{\tan \alpha + \tan \beta}{1 – \tan \alpha \tan \beta} \)
\( \require{AMSsymbols} \begin{align} \tan (\alpha + \beta) &= \displaystyle \frac{\sin (\alpha + \beta)}{\cos (\alpha + \beta)} \\ &= \frac{\sin \alpha \cos \beta + \cos \alpha \sin \beta}{\cos \alpha \cos \beta-\sin \alpha \sin \beta} \\ &= \frac{\displaystyle \frac{\sin \alpha \cos \beta}{\color{red}{\cos \alpha \cos \beta}} + \frac{\displaystyle \cos \alpha \sin \beta}{\color{red}{\cos \alpha \cos \beta}}}{\displaystyle \frac{\cos \alpha \cos \beta}{\color{red}{\cos \alpha \cos \beta}}-\frac{\displaystyle \sin \alpha \sin \beta}{\color{red}{\cos \alpha \cos \beta}}} \\ &= \frac{\displaystyle \frac{\sin \alpha}{\cos \alpha}+\frac{\sin \beta}{\cos \beta}}{1-\displaystyle \frac{\sin \alpha}{\cos \alpha} \times \frac{\sin \beta}{\cos \beta}} \\ &= \displaystyle \frac{\tan \alpha + \tan \beta}{1-\tan \alpha \tan \beta} \\ \therefore \tan (\alpha + \beta) &= \displaystyle \frac{\tan \alpha + \tan \beta}{1-\tan \alpha \tan \beta} \end{align} \)
Example 1
Find the exact value of \( \sin 75^{\circ} \).
\( \begin{align} \sin 75 ^{\circ} &= \sin (45^{\circ} + 30^{\circ}) \\ &= \sin 45^{\circ} \cos 30^{\circ} + \cos 45^{\circ} \sin 30^{\circ} \\ &= \displaystyle \frac{1}{\sqrt{2}} \times \frac{\sqrt{3}}{2} + \frac{1}{\sqrt{2}} \times \frac{1}{2} \\ &= \frac{\sqrt{3}+1}{2 \sqrt{2}} \end{align} \)
Example 2
Find the exact value of \( \cos 75^{\circ} \).
\( \begin{align} \cos 75 ^{\circ} &= \cos (45^{\circ} + 30^{\circ}) \\ &= \cos 45^{\circ} \cos 30^{\circ}-\sin 45^{\circ} \sin 30^{\circ} \\ &= \displaystyle \frac{1}{\sqrt{2}} \times \frac{\sqrt{3}}{2}-\frac{1}{\sqrt{2}} \times \frac{1}{2} \\ &= \frac{\sqrt{3}-1}{2 \sqrt{2}} \end{align} \)
Example 3
Find the exact value of \( \tan 75^{\circ} \).
\( \begin{align} \tan 75 ^{\circ} &= \tan (45^{\circ} + 30^{\circ}) \\ &= \displaystyle \frac{\tan 45^{\circ} + \tan 30^{\circ}}{1-\tan 45^{\circ} \times \tan 30 ^{\circ}} \\ &= \frac{1 + \displaystyle \frac{1}{\sqrt{3}}}{1-\displaystyle 1 \times \frac{1}{\sqrt{3}}} \\ &= \frac{\sqrt{3}+1}{\sqrt{3}-1} \\ &= \frac{(\sqrt{3}+1)^2}{3-1} \\ &= \frac{4+2\sqrt{3}}{2} \\ &= 2+\sqrt{3} \end{align} \)
Algebra Algebraic Fractions Arc Binomial Expansion Capacity Common Difference Common Ratio Differentiation Double-Angle Formula Equation Exponent Exponential Function Factorials Factorise Functions Geometric Sequence Geometric Series Index Laws Inequality Integration Kinematics Length Conversion Logarithm Logarithmic Functions Mass Conversion Mathematical Induction Measurement Perfect Square Perimeter Prime Factorisation Probability Product Rule Proof Quadratic Quadratic Factorise Ratio Rational Functions Sequence Sketching Graphs Surds Time Transformation Trigonometric Functions Trigonometric Properties Volume
