← All topics➕ Compound & Allied AnglesMaths
Trigonometry · Topic 4 of 7

➕ Compound & Allied Angles

2 exam question types, fully solved

A compound angle is an angle written as a sum or difference, like or . An allied angle is a standard angle nudged by a quarter- or half-turn, like or. SSC loves these because, once you spot the pattern, an expression with ugly angles such as and collapses to a value you already know.

The compound-angle formulas

• Sine: — the sign stays the same as the bracket.

• Cosine: — the sign FLIPS (a plus bracket gives a minus inside).

• Tangent: .

The exam habit to build: when you see , do not compute each term — recognise it as , add the angles, and read off the standard value.

Allied-angle reduction: to simplify something like, follow two steps.

• Name change: at and the function swaps to its co-function (, ); at and the name stays the same.

• Sign: decide which quadrant the angle lands in and keep the sign that the original function carries there. So (second quadrant, sine positive), while (second quadrant, cosine negative).

📐Right Triangle ExplorerPick an angle — see exact fraction values
Select angle θ
30°B = √3/2P = 1/2H = 1
sin θP / H1/2
cos θB / H√3/2
tan θP / B1/√3
cosec θH / P2
sec θH / B2/√3
cot θB / P√3

Apply the reduction rule yourself: at the name changes, so . This co-function link is exactly why complementary products such as turn into.

📝Practice Questions

Q1cos40°cos50° − sin40°sin50° = ?

Q2tan20° × tan70° = ?

Q3sin(180° − 30°) = ?

📚

Real exam questions — Compound & Allied Angles

2 question types · 9 solved examples from real SSC papers

Compound-angle and allied-angle questions look intimidating but almost always reduce to a single formula applied backwards, turning odd angles into standard values you already know.

How to solve this type
Memorise three pairs: sin⁡(A±B)=sin⁡Acos⁡B±cos⁡Asin⁡B\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B, cos⁡(A±B)=cos⁡Acos⁡B∓sin⁡Asin⁡B\cos(A\pm B)=\cos A\cos B\mp\sin A\sin B (the sign flips), and tan⁡(A±B)=tan⁡A±tan⁡B1∓tan⁡Atan⁡B\tan(A\pm B)=\dfrac{\tan A\pm\tan B}{1\mp\tan A\tan B}. For allied angles the rule is: at 90∘90^\circ and 270∘270^\circ the function changes to its co-function (sin⁡↔cos⁡\sin\leftrightarrow\cos, tan⁡↔cot⁡\tan\leftrightarrow\cot); at 180∘180^\circ and 360∘360^\circ it stays the same. The sign is fixed by the quadrant the angle lands in. Most PYQs are just one of these formulas read backwards, and the odd angles usually add up to a standard value.

Evaluate cos⁡54∘cos⁡36∘−sin⁡54∘sin⁡36∘\cos 54^\circ\cos 36^\circ-\sin 54^\circ\sin 36^\circ.

A00B12\dfrac12C11D32\dfrac{\sqrt3}{2}

If sin⁡A=35\sin A=\dfrac35 and cos⁡B=513\cos B=\dfrac{5}{13} (both acute), find sin⁡(A+B)\sin(A+B).

A3365\dfrac{33}{65}B5665\dfrac{56}{65}C6365\dfrac{63}{65}D1665\dfrac{16}{65}

If sin⁡P=1213\sin P=\dfrac{12}{13} and cos⁡Q=35\cos Q=\dfrac35 (both acute), find cos⁡(P−Q)\cos(P-Q).

A6365\dfrac{63}{65}B1665\dfrac{16}{65}C3365\dfrac{33}{65}D5665\dfrac{56}{65}

Find the value of tan⁡10∘⋅tan⁡80∘\tan 10^\circ\cdot\tan 80^\circ.

A00B11C3\sqrt3D−1-1

Simplify sin⁡(90∘−A)⋅cos⁡(90∘−A)\sin(90^\circ-A)\cdot\cos(90^\circ-A).

Atan⁡A\tan ABcos⁡2A\cos^2 ACsin⁡2A\sin^2 ADsin⁡Acos⁡A\sin A\cos A