Chapter 3

Trigonometric Functions of Angles

Section 3.2

Trigonometry of Right Triangles

Hypotenuse, Adjacent and Opposite sides of a Triangle

In a right triangle (a triangle with a right angle) the side that does not make up the right angle is called the hypotenuse. For an angle that is not the right angle the other two sides are names in relation to it. The opposite side is a side that makes up the right angle that is across from . The adjacent side is the side that makes up the right angle that also forms the angle .

hypotenuse

hypotenuse

adjacent side

adjacent side

opposite side

opposite side

The Trigonometric Ratios

For any right triangle if we pick a certain angle we can form six different ratios of the lengths of the sides. They are the sine, cosine, tangent, cotangent, secant and cosecant (abbreviated sin, cos, tan, cot, sec, csc respectively).

hypotenuse

opposite side

adjacent side

hypotenuse

sideoppositesin

hypotenuse

sideadjacentcos

sideadjacent

sideoppositetan

sideadjacent

hypotenusesec

sideopposite

sideadjacentcot

sideopposite

hypotenusecsc

To find the trigonometric ratios when the lengths of the sides of a right triangle are know is a matter of identifying which lengths represent the hypotenuse, adjacent and opposite sides. In the triangle below the sides are of length 5, 12 and 13. We want to find the six trigonometric ratios for each of its angles and .

5

12

13

sin

cos

tan

cot

sec

csc

13

5

13

12

12

5

5

12

12

13

5

13

sin

cos

tan

cot

sec

csc

13

5

13

12

12

5

5

12

12

13

5

13

Notice the following are equal: cossin

cottan cscsec

cossin

cottan cscsec

The angles and are called complementary angles (i.e. they sum up to 90). The “co” in cosine, cotangent and cosecant stands for complementary. They refer to the fact that for complementary angles the complementary trigonometric ratios will be equal.

Pythagorean Theorem and Trigonometric Ratios

The Pythagorean Theorem relates the sides of a right triangle so that if you know any two sides of the triangle you can find the remaining one. This is particularly useful in trig since two sides will then determine all six trigonometric ratios.

a

bc

222 cba

Determine the six trigonometric ratios for the right triangle pictured below.

6

7

First we need to determine the length of the remaining side which we will call x and apply the Pythagorean Theorem.

x

222 76 x

sin

cos

tan

cot

sec

csc

7

13

7

6

6

13

13

6

6

7

13

7 49362 x

132 x

13x

sin

cos

tan

cot

sec

csc

7

13

7

6

6

13

13

6

6

7

13

7

Finding Other Trigonometric Ratios by Knowing One

If one of the trigonometric ratios is known it is possible to find the other five trigonometric ratios by constructing a right triangle with an angle and sides corresponding to the ration given. For example, if we know that the sin = ¾ find the other trigonometric ratios.

1. Make a right triangle and label one angle .

2. Make the hypotenuse length 4 and the opposite side length 3.

3. Find the length of the remaining side.

4. Find the other trigonometric ratios.

43

x =

7

7

169

43

2

2

222

x

x

x

x

7

sin

cos

tan

cot

sec

csc

4

3

4

7

7

3

3

7

7

4

3

4

Similar Triangles and Trigonometric Ratios

Triangles that have the exact same angles measures but whose sides can be of different length are called similar triangles. Similar triangles have sides that are proportional. That is to say the sides are just a multiple of each other. Consider the example above where the sides of one triangle are just three times longer than the side of the other triangle.

3

94

5

15

12

sin

cos

tan

cot

sec

csc

5

3

15

9

5

4

15

12

4

3

12

9

3

4

9

12

4

5

12

15

3

5

9

15

Similar Triangles have equal trigonometric ratios!Many of the ideas in trigonometry are based on this concept.

Trigonometric Ratios of Special Angles

45-45-90 Triangles

If you consider a square where each side is of length 1 then the diagonal is of length . 2

1

1

2x

45

45

sin 45

cos 45

tan 45

cot 45

sec 45

csc 45

2

2

2

1

11

1

11

1

21

2

21

2

30-60-90 Triangles

If you consider an equilateral triangle where each side is of length 1 then the perpendicular to the other side is of length .

2

2

2

1

2

3

1

2

1

23

432

412

22

212

1

1

x

x

x

x

23x

60

sin 60

cos 60

tan 60

cot 60

sec 60

csc 60

3

3

3

1

2

3

32

3

2

sin 30

cos 30

tan 30

cot 30

sec 30

csc 30

30

2

3

2

1

3

2

1

2

3

3

3

3

3

1

x

x

x

x

2

2

11

11

2

2

222

3

32

3

2

2

Finding the Length of Sides of Right Triangles

If you know the length of one of the sides and the measure of one of the angles of a right triangle you can find the length of the other sides by using trigonometric ratios.

37

8x

y

Find the values for x and y.

17

68

x

z

Find the values for x and z.

37sin8

37sin8

x

x

37cos8

37cos8

y

y

68cot17

68cot17

x

x

68csc17

68csc17

z

z

x = 4.81452 x = 6.86845

y = 6.38908 z = 18.3351