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Trigonometry Sine Rule

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TrigonometrySine Rule

R

B

q

c

a

p

Mr Porter

C

A

P

Q

r

b

Definition: The Sine Rule

In any triangle ABC

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

R

B

q

c

a

p

C

A

P

Q

r

b

For triangle ABC

For triangle PQR

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

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

When to apply the sine rule:

Is there 2 sides and 2 angles (or more)?

YES, then use the Sine Rule.

When to apply the sine rule:

Is there 2 sides and 2 angles (or more)?

YES, then use the Sine Rule.

C

P

q

14 cm

118°27’

a

x

r

b

8.3 cm

Q

α

R

81°

43°

p

22.4 cm

A

B

c

Write down the sine rule for this triangle

Write down the sine rule for this triangle

Label the triangle.

Substitute P, p, R and r.

We do not need the ‘Q’ ratio.

Label the triangle.

We do not need the ‘C’ ratio.

Rearrange to make sinα the subject.

Substitute A, a, B and b.

Rearrange to make x the subject.

Evaluate RHS

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

Use calculator.

Convert to deg. & min.

Ambiguous Case – Angles (Only)

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

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

C

C

14.5 cm

a

When to apply the sine rule:

Is there 2 sides and 2 angles (or more)?

YES, then use the Sine Rule.

9.8 cm

14.5 cm

b

9.8 cm

9.8 cm

θ

θ

41°

B

θ

41°

A

A

c

A

B

By supplementary angles:

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

Label the triangle.

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

We do not need the ‘C’ ratio.

Substitute A, a, B and b.

Rearrange to make sin θ the subject.

Case 1:

Evaluate RHS

Case 2:

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

Hence, both answers are correct!

Ambiguous Case – Angles (Only)

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

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

Lets check the supplementary angle METHOD.

R

θ

q

When to apply the sine rule:

Is there 2 sides and 2 angles (or more)?

YES, then use the Sine Rule.

17 cm

p

122°

P

Q

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

8.5 cm

r

Write down the sine rule for this triangle

Label the triangle.

We do not need the ‘P’ ratio.

Substitute Q, R, q and r.

Rearrange to make sin θ the subject.

Case 1:

Evaluate RHS

Case 2:

Hence, the ONLY answer is correct!

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

4000 m

L

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?

H

N

0°

N

0°

N

0°

b

50°

60°

When to apply the sine rule: Is there 2 sides and 2 angles (or more)?

YES, then use the Sine Rule.

l

H

h

4 km

L

d

70°

B

(a)

30°

Label the triangle.

50°

60°

B (ship)

30°

Write down the (side) sine rule for this triangle

320°

70°

40°

We do not need the ‘L’ ratio.

Substitute B, H, b and d, (h).

50°

Rearrange to make d the subject.

(b) From the original diagram :

Use basic alternate angles in parallel line, Bearing and

angle sum of a triangle to find all angles with the ∆BHL.

Use calculator.

Bearing of Ship from Lighthouse H:

H = 90°+50°

H = 140°

Re-draw diagram for clarity

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?

T

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

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

3°

82°

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.

h

x

172°

Write down the (side) sine rule for this triangle

5°

8°

B

X

200 m

Y

Use basic angle sum of a triangle, exterior angle of a triangle

and supplementary angles to find all angles with the ∆AYT

And ∆YTB.

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

We do not need the ‘Y’ ratio.

Substitute X, T, t and x.

Rearrange to make x the subject.

Use calculator

Do NOT ROUND OFF!

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?

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

Fence

B

C

20°

A

10 m

10 m

13 m

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

B = 180 – 26° 24’

B = 153° 36’

S

•

Stake

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

Write down the (side) sine rule for this triangle

Write down the (angle) sine rule for this triangle

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

1) The closest point to A along the fence, is point B.

Hence, we need to find distance AB.

We do not need the ‘S’ ratio.

Substitute A, B, a and b.

2) Look at ∆ABS, to use the sine rule, need to find angle ABS or

angle ASB.

Rearrange to make sin B the subject.

Evaluate RHS

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.

In ∆POA,

N

0°

N

0°

N

0°

N

0°

Not to scale!

64°

x

44°

B

P

112°

d

48°

write down the (side) sine rule for this triangle

23°

88°

y

45°

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

92°

A

4 km

45°

20°

We do not need the ‘P’ ratio.

We do not need the ‘A’ ratio.

Substitute P, B, y and d.

Substitute Q, A, q and y.

Q

Rearrange to make d the subject.

Rearrange to make y the subject.

Use basic alternate angles in parallel line, Bearing and

angle sum of a triangle to find all angles in the diagram.

Use calculator

Do NOT ROUND OFF!

Use calculator and y = 3.6274

Do NOT ROUND OFF!

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!]