Download Using Trigonometric Ratios in Real Life

Sect. 9.5 Trigonometric Ratios
Goal 1
Finding Trigonometric Ratios
Goal 2
Using Trigonometric Ratios in
Real Life
Finding Trigonometric Ratios
We can also name the sides in relation
to an acute angle  (Theta)
Opposite side
Adjacent ,
Opposite Side
and Hypotenuse
of a Right Angle
Triangle.

Adjacent side
Finding Trigonometric Ratios
Trigonometric Ratios
• Ratios which compare the
lengths of the sides of a right
triangle
• the common ratios are
tangent, sine, and cosine.
Finding Trigonometric Ratios
Here is an easy way to remember these relationships
for trig functions and the right triangle. Just write
down this mnemonic:
SOH - CAH - TOA
It is pronounced "so - ka - toe - ah".
- The SOH stands for "Sine of an angle is Opposite over
Hypotenuse."
- The CAH stands for "Cosine of an angle is Adjacent
over Hypotenuse."
- The TOA stands for "Tangent of an angle is Opposite
over Adjacent."
Finding Trigonometric Ratios
Example 1.
The sides of a right triangle are in the ratio 3:4:5, as
shown. Name and evaluate three trigonometric
functions of angle 
Solution:
4
sin  
5
3
cos  
5
4
tan  
3
Using Trigonometric Ratios in Real Life
Example 2
5
In a right triangle, sin θ = 13 . Sketch the triangle, place the
ratio numbers, and evaluate the remaining functions of θ.
Finding Trigonometric Ratios
Example 3
Use a calculator to approximate the
sine, cosine and tangent of 54.
Finding Trigonometric Ratios
Trig functions for the 45- 45- 90 Triangle
1
sin 45 

2
1
cos 45 

2
1
Tan 45   1
1
2
2
2
2
45°
2
1
45°
1
Finding Trigonometric Ratios
Trig functions for the 30- 60- 90 Triangle
3
sin 60 
2
1
cos 60 
2
3
Tan 60 
 3
1
1
sin 30 
2
3
cos 30 
2
1
3
A
tan 30 

3
3
60°
2
1
30°
C
3
B
Using Trigonometric Ratios in Real Life
Example 4
In the figure, find sin .

4
7
Using Trigonometric Ratios in Real Life
Example 5:
In right ABC, hypotenuse AB=15 and angle
A=35º. Find leg BC to the nearest tenth.
Using Trigonometric Ratios in Real Life
Example 7.
Indirect measurement.
Trigonometry is used typically to calculate
things that we cannot measure. To measure the
height h of a flagpole, we could measure a
distance of, say, 100 feet from its base. From
that point P, we could then measure the angle
required to sight the top . If that angle turns out
to be 37°, then
Using Trigonometric Ratios in Real Life
Angle of Elevation
The angle of elevation is always measured from the
ground up. Think of it like an elevator that only goes
up. It is always INSIDE the triangle.
In the diagram at the left, x marks the angle of elevation
of the top of the tree as seen from a point on the ground.
You can think of the angle of elevation in relation to the
movement of your eyes. You are looking straight ahead
and you must raise (elevate) your eyes to see the top of
the tree.
Using Trigonometric Ratios in Real Life
Angle of Depression
In the diagram, x marks the angle of depression of a boat at sea from
the top of a lighthouse.
You can think of the angle of depression in relation to the
movement of your eyes. You are standing at the top of the
lighthouse and you are looking straight ahead. You must
lower (depress) your eyes to see the boat in the water.
Using Trigonometric Ratios in Real Life
Example 9:
The angle of elevation between a point on the
ground and the top of a flag pole is 30 degrees. If
the point on the ground is 100 ft from the bottom
of the pole, how tall is the flag pole?
h
30°
100 ft.
Using Trigonometric Ratios in Real Life
Example 10:
Find the height of the tree.
y
tan 31.8 
71
Using Trigonometric Ratios in Real Life
Example 11:
Find the sine, cosine, and
tangent of B. Then find the
measure of B.
3 34
sin B 
34
5 34
cos B 
34
6
tan B 
10
A
2 34
6
C
10
B = 30.96
B
Homework
9.5 10-38 even, 39-42 all