Sine rule: finding a side

The Maths Habit

SOH CAH TOA only works in a right-angled triangle. How could you find a side in a triangle with no right angle?

Split it into two right-angled triangles, or find a rule that works in any triangle

What is the value of b?

  1. b4 = 63

    b = 8

  2. b5 = 147

    b = 10

  3. b0.5 = 93

    b = 1.5

  4. b0.8 = 30.6

    b = 4

Drop the height h from C. Use sine to write h two ways.

ABCbah

Left triangle: h = b × sin A

Right triangle: h = a × sin B

Both are the same height h. What does that tell you?

b × sin A = a × sin B

Divide by sin A and by sin B:  asin A = bsin B

Check: a = 6 cm, A = 50°, B = 70° gives h = 5.64 cm both ways

asin A = bsin B

Which rule?

Two angles and one side Sine rule

Which side is opposite angle A?

Triangle ABC BC

Find the third angle

Angles of 40° and 65° 75°

Angle A is 40°, angle B is 65° and side a is 8 cm. Find side b.

Opposite sides and angles pair up.
bsin 65°=8sin 40°
b = 8 × sin 65°sin 40°
b = 11.3 cm

Angle A is 30°, angle B is 50° and side a is 6 cm. Find side b.

bsin 50°=6sin 30°
b = 6 × sin 50°sin 30°
b = 9.2 cm

Angle A is 40°, angle B is 65° and side a is 8 cm. Find side b.

Opposite sides and angles pair up.
bsin 65°=8sin 40°
b = 8 × sin 65°sin 40°
b = 11.3 cm

Angle A is 72°, angle B is 35° and side a is 10 cm. Find side b.

bsin 35°=10sin 72°
b = 10 × sin 35°sin 72°
b = 6.0 cm

Sine rule: finding a side worksheet, page 1

Answer sheet

  1. Two lighthouses A and B are 20 km apart. A boat is at C. Angle A is 52° and angle B is 61°. How far is the boat from A?

    19.0 km

  2. You know two sides and the angle between them. Will the sine rule find the third side?

    No: use the cosine rule

  1. Find the length x to 1 decimal place.

    x = 11.1 cm
    9 cmx48°67°ABC

  2. Find the length x to 1 decimal place.

    x = 10.3 cm
    6 cmx32°83°ABC

  3. Find the length x to 1 decimal place.

    x = 11.3 cm
    6 cmx30°110°ABC
Exit tickets to print