Trigonometric functions of compound angles cannot be found by simply distributing the trigonometric ratio; for example, \(\sin(A + B) \neq \sin A + \sin B\). Instead, mathematicians use addition formulae to expand and simplify compound expressions such as \((A + B)\) or \((A - B)\). By substituting \(B = A\) into these addition formulae, double-angle formulae for \(2A\) are derived. These fundamental formulae provide algebraic tools to prove complex trigonometric identities, find exact values of non-standard angles without a calculator, and derive half-angle relationships.
A clear branch diagram showing how addition formulae lead directly to double-angle formulae when \(B = A\), and subsequently to half-angle relations when \(A\) is replaced by \(\frac{A}{2}\).
Addition Formulae\(\sin(A \pm B)\), \(\cos(A \pm B)\), \(\tan(A \pm B)\)
Substitute \(B = A\)
Double-Angle Formulae\(\sin 2A\), the 3 forms of \(\cos 2A\), \(\tan 2A\)
Replace \(A\) with \(\frac{A}{2}\)
Half-Angle Relationships\(\sin \frac{A}{2}\), \(\cos \frac{A}{2}\), \(\tan \frac{A}{2}\)
Use addition formulae to expand compound angles, then derive double-angle and half-angle identities.
Angles in the form \((A + B)\) or \((A - B)\) are known as addition angles or compound angles.
\(\sin(A + B) = \sin A \cos B + \cos A \sin B\)\(\sin(A - B) = \sin A \cos B - \cos A \sin B\)
\(\cos(A + B) = \cos A \cos B - \sin A \sin B\)\(\cos(A - B) = \cos A \cos B + \sin A \sin B\)
\(\tan(A + B) = \dfrac{\tan A + \tan B}{1 - \tan A \tan B}\)\(\tan(A - B) = \dfrac{\tan A - \tan B}{1 + \tan A \tan B}\)
\(\sin 2A = 2 \sin A \cos A\)\(\cos 2A = \cos^2 A - \sin^2 A = 2 \cos^2 A - 1 = 1 - 2 \sin^2 A\)\(\tan 2A = \dfrac{2 \tan A}{1 - \tan^2 A}\)
\(\sin \dfrac{A}{2} = \pm\sqrt{\dfrac{1 - \cos A}{2}}\)\(\cos \dfrac{A}{2} = \pm\sqrt{\dfrac{1 + \cos A}{2}}\)\(\tan \dfrac{A}{2} = \dfrac{1 - \cos A}{\sin A}\)
Rewrite the angle as a sum or difference of special angles \(30^\circ, 45^\circ, 60^\circ\).
Apply the appropriate addition or double-angle formula.
Substitute the exact special-angle ratios.
Simplify and rationalise the denominator if needed.
Sine of double angle
Derived by substituting \(B = A\) into \(\sin(A + B)\).
Tangent of double angle
Derived by substituting \(B = A\) into \(\tan(A + B)\).
Answer in your mind, then press "Check Answer".
Find the exact value of \(\cos 75^\circ\) without using a calculator.
Simplify the expression \(\dfrac{2 \tan 75^\circ}{1 - \tan^2 75^\circ}\).
Prove that \(\dfrac{\sin 4x + \sin 2x}{\cos 4x + \cos 2x + 1} = \tan 2x\).
Assuming double angle means doubling the ratio value: \(\sin 2A = 2 \sin A\).
In reality, \(\sin 2A = 2 \sin A \cos A\).
Misinterpreting half-angle indices: assuming \(\cos \frac{A}{2} = \frac{1}{2} \cos A\).
Halving the angle changes the input inside the function, not the amplitude multiplier of the function.
Assuming only one form exists for \(\cos 2A\).
\(\cos 2A\) has three equivalent expressions, which can be chosen to cancel constants like \(+1\) or \(-1\) during proofs.
Objective: Identify addition formulae, double-angle identities and exact values.
1 Which of the following is the correct identity for \(\cos(A + B)\)?
2 What is the exact value of the expression \(\cos^2 165^\circ - \sin^2 165^\circ\)?
3 By substituting \(B\) with \(A\) in the identity for \(\sin(A + B)\), the double-angle formula obtained is \(\sin 2A =\) ________.
4 Using the addition formula for tangent, the expression \(\dfrac{\tan 70^\circ - \tan 10^\circ}{1 + \tan 70^\circ \tan 10^\circ}\) simplifies to \(\tan\) ________\(^\circ\).
Match each trigonometric expression to its equivalent expansion or simplified identity. On a phone, tap a term first, then tap the matching box.
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