TrigonometrySine Rule

Mr Porter

A

B

C

ac

b P Q

R

pq

r

Definition: The Sine RuleIn any triangle ABC

‘The ratio of each side to the sine of the opposite angle is CONSTANT.

A

B

C

ac

b P Q

R

pq

r

For triangle ABC For triangle PQR

Example 1: Use the sine rule to find the value of x correct to 2 decimal places.

C

A B

8.3

cm x

81° 43°

When to apply the sine rule: Is there 2 sides and 2 angles (or more)?YES, then use the Sine Rule.

Write down the sine rule for this triangle

Label the triangle.

ab

c

We do not need the ‘C’ ratio.

Substitute A, a, B and b.

Rearrange to make x the subject.

Use calculator.

Example 2: Find the size of α in ∆PQR in degrees and minutes.

Q

P

R

22.4 cm

14 cm 118°27’

α

When to apply the sine rule: Is there 2 sides and 2 angles (or more)?YES, then use the Sine Rule.

Write down the sine rule for this triangle

Label the triangle.

p

q

r

We do not need the ‘Q’ ratio. Substitute P, p, R and r.

Rearrange to make sinα the subject.

Evaluate RHS

To FIND angle, use sin-1 (..)

Convert to deg. & min.

Ambiguous Case – Angles (Only)

Example 1 : Use the sine rule to find the size of angle θ. C

BA

41°

14.5 cm

9.8

cm

θ

When to apply the sine rule: Is there 2 sides and 2 angles (or more)?YES, then use the Sine Rule.

Write down the sine rule for this triangle to find an angle

Label the triangle.

ab

c

We do not need the ‘C’ ratio. Substitute A, a, B and b.

Rearrange to make sin θ the subject.

Evaluate RHS

To FIND angle, use sin-1 (..)

But, could the triangle be drawn a different way? The answer is YES!

C

BA41°

14.5 cm

9.8

cm

θA

9.8

cm

θ

By supplementary angles:

Which is correct, test the angle sum to 180°, to find the third angle, α.Case 1:

Case 2:

Hence, both answers are correct!

Ambiguous Case – Angles (Only)

Example 1 : Use the sine rule to find the size of angle θ.

QP

R

122°

θ

17 cm

8.5 cm

When to apply the sine rule: Is there 2 sides and 2 angles (or more)?YES, then use the Sine Rule.

Write down the sine rule for this triangle

Label the triangle.

p

q

r

We do not need the ‘P’ ratio. Substitute Q, R, q and r.

Rearrange to make sin θ the subject.

Evaluate RHS

To FIND angle, use sin-1 (..)

But, could the triangle be drawn a different way? The answer is NO!

Lets check the supplementary angle METHOD.

Which is correct, test the angle sum to 180°, to find the third angle, α.

Case 1:

Case 2:

Hence, the ONLY answer is correct!

Example 3: Points L and H are two lighthouses 4 km apart on a dangerous rocky shore. The shoreline (LH) runs east–west. From a ship (B) at sea, the bearing of H is 320° and the bearing of L is 030°.a) Find the distance from the ship (B) to the lighthouse (L), to the nearest metre.b) What is the bearing of the ship (B) from the lighthouse at H? N

0°N0°

N0°

H L4 km

B (ship)

320°

30°

Use basic alternate angles in parallel line, Bearing and angle sum of a triangle to find all angles with the ∆BHL.

30°

60°

50°

50°

40°

70°

Re-draw diagram for clarity

d

H

B

L4000 m

60°50°

70°

When to apply the sine rule: Is there 2 sides and 2 angles (or more)?YES, then use the Sine Rule.

Label the triangle.

l

b

h

We do not need the ‘L’ ratio. Substitute B, H, b and d, (h).

Rearrange to make d the subject.

Use calculator.

Write down the (side) sine rule for this triangle

(a)

(b) From the original diagram :

Bearing of Ship from Lighthouse H:H = 90°+50°H = 140°

Example 4: To measure the height of a hill a surveyor took two angle of elevation measurements from points X and Y, 200 m apart in a straight line. The angle of elevation of the top of the hill from X was 5° and from Y was 8°.What is the height of the hill, correct to the nearest metre?

Hence, the hill is 46 m high (nearest metre).

h

X Y

B

T

200 m

5° 8°

Use basic angle sum of a triangle, exterior angle of a triangle and supplementary angles to find all angles with the ∆AYT And ∆YTB.

82°

172°

3°

When to apply the sine rule: Is there 2 sides and 2 angles (or more)?YES, then use the Sine Rule to find TY= x.

To find ‘h’, we need either length BY or TY!

x

Write down the (side) sine rule for this triangle

We do not need the ‘Y’ ratio. Substitute X, T, t and x.

Rearrange to make x the subject.

Use calculatorDo NOT ROUND OFF!

To find h, use the right angle triangle ratio’s i.e. sin θ.

Example 5: A wooden stake, S, is 13 m from a point, A, on a straight fence. SA makes an angle of 20° with the fence. If a hores is tethered to S by a 10 m rope, where, on the fence, is the nearest point to A at which it can graze?

Fence

• Stake

A

B C

S

20°

10 m10 m

13 m

1) The closest point to A along the fence, is point B.Hence, we need to find distance AB.

2) Look at ∆ABS, to use the sine rule, need to find angle ABS orangle ASB.

Write down the (angle) sine rule for this triangle

Evaluate RHS

To FIND angle, use sin-1 (..)

From the diagram, it is obvious that angle B is Obtuse.

B = 180 – 26° 24’B = 153° 36’

We do not need the ‘S’ ratio. Substitute A, B, a and b.

Rearrange to make sin B the subject.

Then angle ASB = 182 – (153°36’ + 20) = 6° 24’

Now, apply the sine rule to find the length of AB.

Write down the (side) sine rule for this triangle

AB = 3.259 m

Example 6: Q, A and B (in that order) are in a straight line. The bearings of A and B from Q is 020°T. From a point P, 4 km from Q in a direction NW, the bearing of A and B are 112°T and 064°T respectively. Calculate the distance from A to B.

Use basic alternate angles in parallel line, Bearing and angle sum of a triangle to find all angles in the diagram.

23°45°

48°

92°

44°

88°

d112°

N0°

P N0°

A4 km

N0°

Q

N0°

B

45° 20°

64°Not to scale!

x

y

To find ‘d’, we need to work backward using the sine rule,meaning that we must find either x or y first.

[There are several different solution!]

In ∆POA, write down the (side) sine rule for this triangle

We do not need the ‘P’ ratio. Substitute Q, A, q and y.

Rearrange to make y the subject.

Use calculatorDo NOT ROUND OFF!

In ∆PAB, write down the (side) sine rule. For this triangle.

We do not need the ‘A’ ratio. Substitute P, B, y and d.

Rearrange to make d the subject.

Use calculator and y = 3.6274Do NOT ROUND OFF!