1.
The Sine Ratio
Slide 1
In the previous lesson, you used the Pythagorean Theorem to find the measure of
a missing side (of a right triangle).
Over the next few lessons, you will learn different methods for finding the measure
of a side or an angle.
How to label triangles:
•  (theta) – usually used to identify an angle (a capital letter can also be used to
identify an angle)
• adjacent – the side that is adjacent to the angle
• opposite – the side that is opposite to the angle
• hypotenuse – the side that is directly across from the right angle.

2.
Slide 2
When labelling the right triangle, everything depends on which angle
you plan to work with.
Reminder:
• adjacent – the side that is adjacent to the angle
• opposite – the side that is opposite to the angle
• hypotenuse – the side that is directly across from the right angle.
A
B C

A
B C

opposite
opposite

3.
Slide 3
When labelling the right triangle, everything depends on which angle
you plan to work with.
Reminder:
• adjacent – the side that is adjacent to the angle
• opposite – the side that is opposite to the angle
• hypotenuse – the side that is directly across from the right angle.
A
B C

A
B C

adjacent
adjacent

4.
Slide 4
Tip to help remember which side is which.
Straight line across from angle = opposite (P has a straight line as
part of its shape)
Curved line from angle = adjacent (J is a curved letter)
A
B C

A
B C

adjacent
opposite
opposite
adjacent

5.
Slide 5
Calculator Check …
When working with the trigonometric ratios: sine, cosine and tangent,
you must make sure your calculator is set to Degree Mode.
Quick Check:
sin 30 = 0.5
If you do not get this as an answer,
then your calculator is not in
degree mode.

6.
Slide 6
The Sine Ratio
Formal definition: in a right triangle, the ratio of the length of the side
opposite a given angle to the length of the hypotenuse
A
B C

sin
opp
hyp
 
adjacent
opposite

7.
Example 1:
PQR is shown below. Using the SINE ratio, find the measure of x
(round to 1 decimal place).
Slide 7
x
67
120

8.
Example 1:
PQR is shown below. Using the SINE ratio, find the measure of x
(round to 1 decimal place).
Slide 8
x
67
120
Label the triangle.
Identify the formula.
adjacent
opposite
hypotenuse
sin
opp
hyp
 

9.
Example 1:
PQR is shown below. Using the SINE ratio, find the measure of x
(round to 1 decimal place).
Slide 9
x
67
120
Label the triangle.
Identify the formula.
adjacent
opposite
hypotenuse
sin
opp
hyp
 
sin 67
120
x

0.9205
120
x

*Recommendation:
Use 4 decimal places
 0.9205
1
120 1 0
20
2
x 
  
 
110.46
110.5
x
x

 

10.
Example 2:
PQR is shown below. Using the SINE ratio, find the measure of x
(round to 1 decimal place).
Slide 10
5
23
x

11.
Example 2:
PQR is shown below. Using the SINE ratio, find the measure of x
(round to 1 decimal place).
Slide 11
5
23
x
Label the triangle.
Identify the formula.
sin
opp
hyp
 
adjacent
opposite
hypotenuse
5
sin 23
x

5
0.3907
x

*Recommendation:
Use 4 decimal places
 
5
0.3907
x
x x
 
  
 
0.3907 5x 
0.3907
0.3907
07
5
0.39
x

12.7975
12.8
x
x



12.
Slide 12
New Terms:
Angle of Elevation
Angle of Depression
When you look up at an airplane flying
overhead, the angle between the
horizontal and your line of sight is called
an angle of elevation.
When you look down from a cliff to a
boat passing by, the angle between the
horizontal and your line of sight is called
an angle of depression.

13.
Example 3:
The angle of elevation of Sandra’s kite string is 70. If she has
let out 55 feet of string, and is holding the string 6 feet above the
ground, how high is the kite?
Slide 13

14.
Example 3:
The angle of elevation of Sandra’s kite string is 70. If she has
let out 55 feet of string, and is holding the string 6 feet above the
ground, how high is the kite?
Slide 14
Label the triangle.
Identify the formula.
sin
opp
hyp
 
adjacent
oppositehypotenuse

15.
Example 3:
The angle of elevation of Sandra’s kite string is 70. If she has
let out 55 feet of string, and is holding the string 6 feet above the
ground, how high is the kite?
Slide 15
Label the triangle.
Identify the formula.
sin
opp
hyp
 
adjacent
oppositehypotenuse
sin 70
55
h

0.9397
55
h

 0.9397
5
55 55
5
h 
  
 
51.6835
51.7
51.7
57.7
6
h
h
h
h feet




 *Held 6 feet
above ground