Geometry

Sine rule

Match each side with its opposite angle.

Predict, test and explain

Step 1 of 3

Predict

A triangle has A = 30° and B = 90°. Which side is longest?

asin⁡A=bsin⁡B=csin⁡C\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}
A=50∘A = 50^\circ
B=65∘B = 65^\circ
C=65∘C = 65^\circ
a≈5.07a \approx 5.07
b=6b = 6
c=6c = 6

Scroll the diagram sideways to see every label.

asin⁡A≈6.62\frac{a}{\sin A} \approx 6.62bsin⁡B≈6.62\frac{b}{\sin B} \approx 6.62csin⁡C≈6.62\frac{c}{\sin C} \approx 6.62

InvestigateWhich side faces the biggest angle?

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Hints, self-check and connections
Try a self-check

Fixed values, separate from the diagram controls.

In triangle ABC, each lower-case side is opposite its matching angle. Find side b exactly, in cm.

a=6 cm,A=30∘,B=60∘a=6\,\text{cm},\qquad A=30^\circ,\qquad B=60^\circ

Hint

Pair each side with its opposite angle and use the sine rule.

Reveal answer

Multiply 6 by the ratio of the two sine values.

b=6sin⁡60∘sin⁡30∘=63 cmb=\frac{6\sin60^\circ}{\sin30^\circ}=6\sqrt3\,\text{cm}

Build an explanation

Hint 1

Pair each side with the angle opposite it.

Hint 2

Compare side length divided by the sine of its opposite angle.

Hint 3

The sine rule is a/sin A = b/sin B = c/sin C.

Worked example

This example uses fixed values, separate from the diagram controls.

A triangle has A = 30°, C = 90° and c = 6 cm. Find a.

  1. a faces 30°; c faces 90°.
  2. a/sin 30° = 6/sin 90°.
  3. Multiply by sin 30°: a = 6 × 0.5/1.

Answer: 3 cm.

Connect this idea

Watch out: Use opposite pairs, not a side and a nearby angle; set your calculator to degrees.