The document provides instructions on using trigonometric functions to find missing angles or sides of right triangles. It begins with reminders of trigonometric definitions and ratios. Examples are then given to demonstrate finding a missing side using the Pythagorean theorem or a trigonometric ratio, and finding a missing angle using an inverse trigonometric function. Tips are provided on determining which method to use based on the information given in a problem.

1. Finding Angles
Slide 1
In the past few lessons, you have learned how to use sin, cos and tan to find a
length of a side in a right triangle.
In this lesson, you will learn how to use sin, cos and tan to find a missing angle.
Reminders:
•  (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.
• degree mode – remember to set your calculator to degree mode
– quick check: sin 30 = 0.5

2. Slide 2
cos
adj
yph
 sin
opp
yph
  tan
opp
dja
 
soh cah toa

3. Slide 3
Find the following trig values to 4 decimal places:
sin 30 = sin 120 =
cos 30 = cos 120 =
tan 30 = tan 120 = -1.73215

4. sin 30 = sin 120 =
cos 30 = cos 120 =
tan 30 = tan 120 =
0.5000
0.8660
0.5774
-0.5000
0.8660
Slide 4
Find the following trig values to 4 decimal places:
-1.73215

5. Slide 5
In the previous lessons, you calculated the sine, cosine or tangent of an
angle. Example: sin 85 = 0.9962
It is also possible to calculate the angle when you are given a value.
Example:
sin A = 0.7071
A = 45
On your calculator, you will use the inverse sine, cosine or tangent
buttons to calculate the angle.

6. Slide 6
sin A = 0.7071
On your calculator,
• enter 0.7071
• press the 2nd button (or shift)
• press the sin button
You should get an answer of 44.99  A = 45
Is your calculator
in degree mode?

7. Slide 7
Steps to calculate the angle:
• enter the value
• press the 2nd button (or shift)
• press the sin-1, cos-1, or tan-1 button
Try these:
Find A: sin A = 0.9962
You should get an answer of A = 85
Is your calculator
in degree mode?
Find A: tan A = 28.6363
You should get an answer of A = 88
Find A: cos A = 0.5878
You should get an answer of A = 54

8. sin A = 0.91847 sin A = 0.8660
cos A = 0.62159 cos A = -0.7071
tan A = 1.7321 tan A = 0.2679
Slide 8
Using the inverse trig functions, find the measure of the angle (to 1
decimal place):

9. A = 66.7
A = 51.6
A = 60 A = 15
A = 135
A = 60sin A = 0.91847 sin A = 0.8660
cos A = 0.62159 cos A = -0.7071
tan A = 1.7321 tan A = 0.2679
Slide 9
Using the inverse trig functions, find the measure of the angle (to 1
decimal place):

10. If you need to find a missing
side and you know…
• at least one side
• at least one angle
(not the right angle)
Use SIN, COS, or TAN
Slide 10
Tips for Solving Triangle Problems
If you need to find the measure of an
angle and you know ….
• two or more sides
and
• the angles are unknown
Use the inverse of SIN, COS, or TAN
(2nd / Shift button is used before SIN, COS, or TAN)
If you need to find a missing
side and you know…
• two sides
Use Pythagorean Theorem
a2 + b2 = c2

11. Example 1:
Find the value of x (to two decimal places).
Slide 11
2.71
x
4.29

12. Example 1:
Find the value of x (to two decimal places).
Slide 12
2.71
x
4.29
What do you need to find?
What do you know?
What will you use to find x?

13. Example 1:
Find the value of x (to two decimal places).
Slide 13
2.71
x
4.29
What do you need to find?
What do you know?
What will you use to find x?
Missing side
Two sides
Pythagorean Theorem
2 2 2
a b c 

14. Example 1:
Find the value of x (to two decimal places).
Slide 14
2.71
x
4.29
What do you need to find?
What do you know?
What will you use to find x?
Missing side
Two sides
Pythagorean Theorem
2 2 2
a b c 
Identify the sides.
   
2 2 2
2.71 4.29 c 
2
7.3441 18.4041 c 
2
7.3441 18.4041 c 
a
b
c
2
25.7482 c
2
25.7482 c
5.0743
5.07
c
c


Reminder – lengths are always
positive so the  sign is not required.

15. Slide 15
57.73
x
35
Example 2:
Find the value of x (to two decimal places).

16. Slide 16
What do you need to find?
What do you know?
What do you want to know?
57.73
x
35
Example 2:
Find the value of x (to two decimal places).
adjacent
opposite
hypotenuse
What will you use to find x?
Label the triangle

17. Slide 17
What do you need to find?
What do you know?
What do you want to know?
57.73
x
35
Example 2:
Find the value of x (to two decimal places).
Missing side
Angle and a side
Know HYPotenuse
adjacent
opposite
hypotenuse
What will you use to find x?
OPPosite
sin
opp
hyp
 
Label the triangle

18. Slide 18
What do you need to find?
What do you know?
What do you want to know?
57.73
x
35
Example 2:
Find the value of x (to two decimal places).
Missing side
Angle and a side
Know HYPotenuse
adjacent
opposite
hypotenuse
What will you use to find x?
OPPosite
sin
opp
hyp
 
Label the triangle
sin35
57.73
x

0.5736
57.73
x

 0.5736
57.
57.73 57
73
.73
x 
  
 
33.1139
33.11
x
x



19. Example 3:
Find the value of x (to two decimal places).
Slide 19
6.48
10.87
x

20. Example 3:
Find the value of x (to two decimal places).
Slide 20
What do you need to find?
What do you know?
What will you use to find x?
6.48
10.87
x
adjacent
opposite
hypotenuse
Label the triangle

21. Example 3:
Find the value of x (to two decimal places).
Slide 21
What do you need to find?
What do you know?
What will you use to find x?
6.48
10.87
x
adjacent
opposite
hypotenuse
Label the triangle
Missing angle
Two sides: ADJacent and OPPosite
Inverse of tan: tan-1
tan
opp
adj
 

22. Example 3:
Find the value of x (to two decimal places).
Slide 22
What do you need to find?
What do you know?
What will you use to find x?
6.48
10.87
x
adjacent
opposite
hypotenuse
Label the triangle
Missing angle
Two sides: ADJacent and OPPosite
Inverse of tan: tan-1
tan
opp
adj
 
10.87
tan
6.48
x 
tan 1.6775x 
59.1998
59.20
x
x

 
Steps to calculate the angle:
• enter the value
• press the 2nd button (or shift)
• press the tan-1 button

23. Example 4:
A pirate’s treasure is buried at 100 m directly below the surface of the earth. A
hole is to be drilled in order to get at the treasure. Drilling begins directly above
the treasure but the drilling is off the vertical line by 5. By how much will the hole
miss the treasure when the hole is at the same depth as the treasure?
Slide 23

24. Example 4:
A pirate’s treasure is buried at 100 m directly below the surface of the earth. A
hole is to be drilled in order to get at the treasure. Drilling begins directly above
the treasure but the drilling is off the vertical line by 5. By how much will the hole
miss the treasure when the hole is at the same depth as the treasure?
Slide 24
100 m
5°
x treasure
Draw a diagram

25. Example 4:
A pirate’s treasure is buried at 100 m directly below the surface of the earth. A
hole is to be drilled in order to get at the treasure. Drilling begins directly above
the treasure but the drilling is off the vertical line by 5. By how much will the hole
miss the treasure when the hole is at the same depth as the treasure?
Slide 25
What do you need to find?
What do you know?
What will you use to find x?
Label the triangle
100 m
5°
x treasure
adjacent
opposite
hypotenuse

26. Example 4:
A pirate’s treasure is buried at 100 m directly below the surface of the earth. A
hole is to be drilled in order to get at the treasure. Drilling begins directly above
the treasure but the drilling is off the vertical line by 5. By how much will the hole
miss the treasure when the hole is at the same depth as the treasure?
Slide 26
What do you need to find?
What do you know?
What will you use to find x?
Label the triangle
100 m
5°
x treasure
adjacent
opposite
hypotenuse
Missing side (OPPosite)
Angle and ADJacent
Inverse of tan: tan-1
tan
opp
adj
 

27. Example 4:
A pirate’s treasure is buried at 100 m directly below the surface of the earth. A
hole is to be drilled in order to get at the treasure. Drilling begins directly above
the treasure but the drilling is off the vertical line by 5. By how much will the hole
miss the treasure when the hole is at the same depth as the treasure?
Slide 27
What do you need to find?
What do you know?
What will you use to find x?
Label the triangle
100 m
5°
x treasure
adjacent
opposite
hypotenuse
Missing side (OPPosite)
Angle and ADJacent
Inverse of tan: tan-1
tan
opp
adj
 
tan5
100
x

0.0875
100
x

 0.0875
1
100 1 0
00
0
x 
  
 
8.75x m
Answer:
The drill will miss the
treasure by 8.75m.