Download Sine and Cosine Ratios

LESSON 3: THE SINE AND COSINE RATIOS
Learning Outcome: Learn to develop and apply the sine and cosine ratios
to determine angle measures.
Key Math Learnings: In a right triangle, there are two trigonometric ratios
that relate the opposite side to the hypotenuse and the adjacent side to the
hypotenuse.
Making Connections:
Has anyone ever travelled on a train?
If not, while travelling by car, have you ever noticed a train going through
tunnels?
Why do you think tunnels were being constructed for trains?
Now if tunnels were not being used, the trains would have to travel along
an incline during transit. Here is a diagram of the track before the tunnels
were constructed.
Point B
6.6 km
297 m
Point A
How would you determine the angle of inclination of the track?
1. Drawing a scale diagram (too big to construct)
2. use Pythagorean theorem to determine the length of the adjacent side,
then use the tangent ratio
In a right triangle, the ratios that relate each leg to the hypotenuse depend
only on the measure of the acute angle, and not on the size of the triangle.
These ratios are called the sine ratio and the cosine ratio.
sin 𝐴 =
cos 𝐴 =
𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑠𝑖𝑑𝑒 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 ∠𝐴
𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
C
𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑠𝑖𝑑𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑜 ∠𝐴
Opposite
∠A
hypotenuse
𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
B
Adjacent to ∠
A
A
The tangent, sine and cosine are called primary trigonometric ratios. The
trigonometry means “three angle measure.”
Ex. Determine sin 32 and cos 32, rounded to 4 decimal places and explain
the meaning of the results. Label the triangle first.
Sin(32) = 0.5299
Cos(32) = 0.8480
32˚
Sin 32 = 0.5299 means that the side opposite
the 32 angle is 0.5299 times as long as the
hypotenuse
cos 32 = 0.8480 means that the side
adjacent to the 32 angle is 0.8480 as
long as the hypotenuse
B
Ex. Write each trigonometric ratio
a. sin A
b. cos A
c. sin B
d. cos B
5
4
A
3
C
Ex. Determine the measures of ∠K and ∠M to the nearest tenth of a
degree.
8
K
M
3
N
Second way to find ∠M?
Ex. In the World Cup Downhill held at Panorama Mountain Village in BC,
the skiers raced 3514m down the mountain. If the vertical height of the
course was 984m, determine the average angle of the ski course with the
ground.
3514m
984m
𝜃
Assignment: pg. 95-96 #1-13, 15,