Trigonometry Sine Rule Mr Porter A B C a c b PQ R p q r.
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Transcript of Trigonometry Sine Rule Mr Porter A B C a c b PQ R p q r.
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!