Download Sine and Cosine Ratios

UNIT 6 – TRIGONOMETRY OF RIGHT TRIANGLES
Lesson 6 – Sine and Cosine Ratios
Labelling – Label the sides of each triangle.
A
θ
θ
B
C
Sine Ratio – the ratio of the opposite side to the hypotenuse
𝑠𝑖𝑛 𝜃 =
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
Cosine Ratio – the ratio of the adjacent side to the hypotenuse
𝑐𝑜𝑠 𝜃 =
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
From ∠𝐴
From ∠𝐶
sin A =
sin C =
3 cm
cos A =
cos C =
B
A
5 cm
C
4 cm
Mneumonic:
S O H
𝑠𝑖𝑛 𝜃 =
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
C A H
,
𝑐𝑜𝑠 𝜃 =
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
T O A
,
𝑡𝑎𝑛 𝜃 =
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
D) Examples
Example 1: Determine sin A and cos A, correct to the nearest thousandth.
B
C
5 cm
10 cm
A
Example 2: Evaluate to the nearest thousandth.
a) sin 54° 
b) sin 19° 
c) cos 89° 
d) cos 6° 
Example 3: Calculate ∠𝐴, to the nearest degree.
a) sin A =
1
3
b) sin A = 0.631
c) cos A = 0.217
d) cos A = 0.217
Applications of Sine and Cosine Ratios
Example 4: A rope attached to a kite is 10 m long. The boy holding the rope estimates that the
angle between the rope and the ground (angle of elevation) is 35°. Calculate the height of the kite.
Example 5: A ladder leaning against a wall makes an angle of 53° with the ground (angle of
elevation). If the foot of the ladder is 1.5 m from the wall, calculate the length of the ladder.
Example 6: Solve the triangle.
NOTE: To solve the triangle means to find all the unknown sides and all the unknown angles.
a)
b) A
A
B
61°
23 cm
9 cm
B
18 cm
C
C