TMUA Triangle Trigonometry and Radians

Triangle rules

• Sine rule: $\dfrac{a}{\sin A}=\dfrac{b}{\sin B}$.
• Cosine rule: $a^2=b^2+c^2-2bc\cos A$.
• Area $=\tfrac12ab\sin C$.

Radians

$\pi$ radians is $180^\circ$. For an angle $\theta$ in radians, arc length is $r\theta$ and sector area is $\tfrac12r^2\theta$.

Segments

A segment is a sector minus a triangle: area $=\tfrac12r^2(\theta-\sin\theta)$.

Exact values and checks

Use exact values: $\sin30^\circ=\tfrac12$, $\cos60^\circ=\tfrac12$, $\sin45^\circ=\tfrac{\sqrt2}2$, $\sin60^\circ=\tfrac{\sqrt3}2$ and $\tan45^\circ=1$. Remember $\tfrac\pi6=30^\circ$, $\tfrac\pi4=45^\circ$ and $\tfrac\pi3=60^\circ$.

Sketch every triangle. In 3D, find a right-angled triangle inside the solid first, then use SOH CAH TOA or Pythagoras.

The ambiguous case arises when you know two sides and an angle that is not between them. The sine rule can then give two possible angles, $\theta$ and $180^\circ-\theta$, and you must test whether both fit in a triangle.

For a sector, always check that the angle is in radians before using $r\theta$ or $\tfrac12r^2\theta$, and give answers in terms of $\pi$ where the question allows.

Worked examples

Common mistakes

• Using the sector formulas with an angle in degrees.
• Using the cosine rule with the angle not between the two known sides.
• Forgetting the ambiguous case when using the sine rule to find an angle.
• Giving the sector area for a segment question.

Test yourself on this topic with our free practice questions.