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MathematicsGCSE, IGCSE & A Level

What is trigonometry?

Trigonometry relates the angles of a triangle to the lengths of its sides using the sine, cosine and tangent ratios.

Trigonometry explained

In right-angled triangles, SOHCAHTOA links each ratio to opposite, adjacent and hypotenuse. In any triangle, the sine and cosine rules take over.
At A Level the ratios become functions with periodic graphs, and identities such as sin⁡2θ+cos⁡2θ=1\sin^{2}\theta + \cos^{2}\theta = 1 let you solve trigonometric equations.

Key formula

asin⁡A=bsin⁡B=csin⁡C\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}

Worked example

In a triangle with a=7a = 7, A=40∘A = 40^{\circ} and B=60∘B = 60^{\circ}: b=7sin⁡60∘sin⁡40∘≈9.4b = \frac{7\sin 60^{\circ}}{\sin 40^{\circ}} \approx 9.4.

Examiner tip

Check your calculator is in degrees for GCSE work and in radians for A Level calculus questions — a whole question can be lost to the wrong mode.

Trigonometry: common questions

When do I use the sine rule and when the cosine rule?
Use the sine rule with a matching side-angle pair, and the cosine rule when you know three sides, or two sides and the angle between them.
What is the ambiguous case?
When the sine rule gives an angle, a second obtuse solution 180° minus that angle may also fit the triangle — always check whether the context allows it.

Related terms

Pythagoras' theorem

Pythagoras' theorem states that in a right-angled triangle the square of the hypotenuse equals the sum of the squares of the other two sides.

Radian

A radian is the angle subtended at the centre of a circle by an arc equal in length to the radius; 2π radians make a full turn.

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