1. Introduction to Trigonometry
Right Triangle Trigonometry

2. Introduction
What special theorem do you already know that applies to a right triangle?
Pythagorean Theorem: a2 + b2 = c2 c a b

3. Introduction
Trigonometry is a branch of mathematics that uses right triangles to help you solve problems.
Trig is useful to surveyors, engineers, navigators, and machinists (and others too.)

5. Finding Trigonometric Ratios
A trigonometric ratio is a ratio of the lengths of two sides of a right triangle.
The word trigonometry is derived from the ancient Greek language and means measurement of triangles.
The three basic trigonometric ratios are sine, cosine, and tangent, which are abbreviated as sin, cos, and tan, respectively.

6. Topic 1
Before we can understand the trigonometric ratios we need to know how to label Right Triangles.

7. Labeling Right Triangles
The most important skill you need right now is the ability to correctly label the sides of a right triangle.
The names of the sides are:
the hypotenuse
the opposite side
the adjacent side

8. Labeling Right Triangles
The hypotenuse is easy to locate because it is always found across from the right angle.
Since this side is across from the right angle, this must be the hypotenuse.
Here is the right angle...

9. Labeling Right Triangles
Before you label the other two sides you must have a reference angle selected.
It can be either of the two acute angles.
In the triangle below, let’s pick angle B as the reference angle. A B C
This will be our reference angle...

10. Labeling Right Triangles
Remember, angle B is our reference angle.
The hypotenuse is side BC because it is across from the right angle. A
B (ref. angle) C
hypotenuse

11. Labeling Right Triangles
Side AC is across from our reference angle B. So it is labeled: opposite. A
B (ref. angle) C
hypotenuse
opposite

12. Labeling Right Triangles
Adjacent means beside or next to
The only side unnamed is side AB. This must be the adjacent side. A
B (ref. angle) C
adjacent
hypotenuse
opposite

13. Labeling Right Triangles
Let’s put it all together.
Given that angle B is the reference angle, here is how you must label the triangle: A
B (ref. angle) C
hypotenuse
adjacent
opposite

14. Labeling Right Triangles
Given the same triangle, how would the sides be labeled if angle C were the reference angle?
Will there be any difference?

15. Labeling Right Triangles
Angle C is now the reference angle.
Side BC is still the hypotenuse since it is across from the right angle. B
hypotenuse
C (ref. angle) A

16. Labeling Right Triangles
However, side AB is now the side opposite since it is across from angle C. B
opposite
hypotenuse
C (ref. angle) A

17. Labeling Right Triangles
That leaves side AC to be labeled as the adjacent side. B
hypotenuse
opposite
C (ref. angle) A
adjacent

18. Labeling Right Triangles
Let’s put it all together.
Given that angle C is the reference angle, here is how you must label the triangle: A B
C (ref. angle)
hypotenuse
opposite
adjacent

19. Labeling Practice
Given that angle X is the reference angle, label all three sides of triangle WXY.
Do this on your own. Click to see the answers when you are ready. W X Y

20. Labeling Practice How did you do? Click to try another one... W X Y
adjacent
opposite
hypotenuse

21. Labeling Practice
Given that angle R is the reference angle, label the triangle’s sides.
Click to see the correct answers. R S T

22. Labeling Practice The answers are shown below: R T S hypotenuse
adjacent
opposite

23. Finding Trigonometric Ratios
A trigonometric ratio is a ratio of the lengths of two sides of a right triangle.
TRIGONOMETRIC RATIOS
Let be a right triangle. The sine, the cosine, and the tangent of the acute angle A are defined as follows.
ABC
side opposite A
hypotenuse = o h B
sin A =
hypotenuse
side
opposite A h o
side adjacent A
hypotenuse = a h
cos A = A C a
side opposite A
side adjacent to A = o a
tan A =
side adjacent to A
The value of the trigonometric ratio depends only on the measure of the acute angle, not on the particular right triangle that is used to compute the value.

24. Finding Trigonometric Ratios
Compare the sine, the cosine, and the tangent ratios for A in each triangle below. A B C
8.5 4
7.5
SOLUTION
These triangles were created so that  A has the same
measurement in both triangles. A B C 17 8 15
Large triangle
Small triangle
sin A =
opposite
hypotenuse  8 17  4
8.5
cos A =
adjacent
hypotenuse  15 17 
7.5
8.5
tan A =
opposite
adjacent  4
7.5  8 15
Trigonometric ratios are frequently expressed as decimal approximations.
The ratios of the sides for a certain angle size stays constant. We can use this to help us find missing sides and missing angles.

25. How do I remember these?

26. Finding Trigonometric Ratios
Find the sine, the cosine, and the tangent of the indicated angle. R T S 5 13 12 S
SOLUTION
The length of the hypotenuse is 13. For S, the length of the opposite side is 5, and the length of the adjacent side is 12.
opp.
hyp. = = 5 13
sin S  R T S 5 12 13
adj.
hyp. = = 12 13
hyp.
cos S 
opp.
opp.
adj. = = 5 12
adj.
tan S 

27. Finding Trigonometric Ratios
Find the sine, the cosine, and the tangent of the indicated angle. R T S 5 13 12 R
SOLUTION
The length of the hypotenuse is 13. For R, the length of the opposite side is 12, and the length of the adjacent side is 5.
opp.
hyp. = = 12 13
sin R  R T S 5 12 13
adj.
hyp. = = 5 13
hyp.
cos R 
adj.
opp.
adj. = = 12 5
opp.
tan R
= 2.4

28. Trigonometric Ratios for 45º
Find the sine, the cosine, and the tangent of 45º.
SOLUTION
Because all such triangles are similar, you can make calculations simple by choosing 1 as the length of each leg.
From the 45º-45º-90º Triangle Theorem, it follows that the length of the hypotenuse is
opp.
hyp. = = 1 2 = 2
sin 45º 
45º 2
hyp. 1
adj.
hyp. = = 1 2 = 2
cos 45º  1
opp.
adj. = = 1
tan 45º
= 1

29. Trigonometric Ratios for 30º
Find the sine, the cosine, and the tangent of 30º.
SOLUTION
To make the calculations simple, you can choose 1 as the length of the shorter leg.
From the 30º-60º-90º Triangle Theorem, it follows that the length of the longer leg is 3 and the length of the hypotenuse is 2.
opp.
hyp. = = 1 2
sin 30º
= 0.5
30º 2
adj.
hyp. = 3 2 =
cos 30º  1
opp.
adj. = = 1 3 3 =
tan 30º  3

30. Sample keystroke sequences Sample calculator display Rounded
Using a Calculator
You can use a calculator to approximate the sine, the cosine, and the tangent of 74º. Make sure your calculator is in degree mode. The table shows some sample keystroke sequences accepted by most calculators.
Sample keystroke sequences
Sample calculator display
Rounded
approximation 74 or
sin
Enter
0.9613 74 or
cos
Enter
0.2756 74 or
tan
Enter
3.4874

31. Finding Trigonometric Ratios
The sine or cosine of an acute angle is always less than 1.
The reason is that these trigonometric ratios involve the ratio of a leg of a right triangle to the hypotenuse.
The length of a leg of a right triangle is always less than the length of its hypotenuse, so the ratio of these lengths is always less than one.
Because the tangent of an acute angle involves the ratio of one leg to another leg, the tangent of an angle can be less than 1, equal to 1, or greater than 1.

32. Finding Trigonometric Ratios
The sine or cosine of an acute angle is always less than 1.
The reason is that these trigonometric ratios involve the ratio of a leg of a right triangle to the hypotenuse.
The length of a leg of a right triangle is always less than the length of its hypotenuse, so the ratio of these lengths is always less than one.
Because the tangent of an acute angle involves the ratio of one leg to another leg, the tangent of an angle can be less than 1, equal to 1, or greater than 1.
TRIGONOMETRIC IDENTITIES
A trigonometric identity is an equation involving trigonometric ratios that is true for all acute angles. The following are two examples of identities:
(sin A) 2 + (cos A) 2 = 1 A B C c a b
tan A =
sin A
cos A

33. Using Trigonometric Ratios in Real Life
Suppose you stand and look up at a point in the distance, such as the top of the tree. The angle that your line of sight makes with a line drawn horizontally is called the angle of elevation.

34. The tree is about 75 feet tall.
Indirect Measurement
FORESTRY You are measuring the height of a Sitka spruce tree in Alaska. You stand 45 feet from the base of a tree. You measure the angle of elevation from a point on the ground to the top of the tree to be 59°. To estimate the height of the tree, you can write a trigonometric ratio that involves the height h and the known length of 45 feet.
tan 59° =
opposite
adjacent
Write ratio.
tan 59° =
opposite
adjacent h 45
Substitute.
45 tan 59° = h
Multiply each side by 45.
45(1.6643)  h
Use a calculator or table to find tan 59°.
74.9  h
Simplify.
The tree is about 75 feet tall.

35. A person travels 152 feet on the escalator stairs.
Estimating a Distance
ESCALATORS The escalator at the Wilshire/Vermont Metro Rail Station in Los Angeles rises 76 feet at a 30° angle. To find the distance d a person travels on the escalator stairs, you can write a trigonometric ratio that involves the hypotenuse and the known leg length of 76 feet.
sin 30° =
opposite
hypotenuse
Write ratio for sine of 30°.
sin 30° =
opposite
hypotenuse 76 d
Substitute.
30°
76 ft d
d sin 30° = 76
Multiply each side by d.
d = 76
sin 30°
Divide each side by sin 30°.
d = 76
0.5
Substitute 0.5 for sin 30°.
d = 152
Simplify.
A person travels 152 feet on the escalator stairs.

36. Homework
Pg. 469 #3,4,13
Pg. 477 # 3,4,6,7,8