Section 6.1

Trigonometric Functions of Acute

Angles

Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.

Objectives

Determine the six trigonometric ratios for a given acute angle of a right triangle.

Determine the trigonometric function values of 30º, 45º, and 60º.

Using a calculator, find function values for any acute angle, and given a function value of an acute angle, find the angle.

Given the function values of an acute angle, find the function values of its complement.

Right Triangles and Acute AnglesAn acute angle is an angle with measure greater than 0º and less than 90º.

Greek letters such as (alpha), (beta), (gamma), (theta), and (phi) are often used to denote an angle.

Side opposite

Side adjacent to

Hypotenuse

We label the sides with respect to angles. The hypotenuse is opposite the right angle. There is the side opposite and the side adjacent to .

Trigonometric Ratios

The lengths of the sides of a right triangle are used to define the six trigonometric ratios:

sine (sin)

cosine (cos)

tangent (tan)

Side opposite

Side adjacent to

Hypotenuse

cosecant (csc)

secant (sec)

cotangent (cot)

Trigonometric Function Values of an Acute Angle Let be an acute angle of a right triangle. Then the six trigonometric functions of are as follows:

sin side opposite

hypotenuse

cos side adjacent to

hypotenuse

tan side opposite

side adjacent to

csc hypotenuse

side opposite

sec hypotenuse

side adjacent to

cot side adjacent to side opposite

Example

In the triangle shown, find the six trigonometric function values of (a) and (b) .

a) sin opp

hyp

12

13

13

5

12

cos adj

hyp

5

13

tan opp

adj

12

5

csc hyp

opp

13

12

sec hyp

adj

13

5

cot adj

opp

5

12

Example

In the triangle shown, find the six trigonometric function values of (a) and (b) .

a) sin opp

hyp

5

13

13

5

12

cos adj

hyp

12

13

tan opp

adj

5

12

csc hyp

opp

13

5

sec hyp

adj

13

12

cot adj

opp

12

5

Reciprocal Functions

Note that there is a reciprocal relationship between pairs of the trigonometric functions.

csc 1

sin1

seccos

cot 1

tan

Example

Given that sin 4

5, cos

3

5, and tan

4

3,

Solution:

csc 1

sin

14

5

5

4

sec 1

cos

13

5

find csc , sec , and cot .

5

3

cot 1

tan

14

3

3

4

Example

If and is an acute angle, find the other five trigonometric function values of .

sin 6

7

Solution:

6

7

opp

hyp

Use the definition of the sine function that the ratio

and draw a right triangle.

7

a

6

Use the Pythagorean equation to find a. a2 b2 c2

a2 62 72

a2 36 49

a2 49 36 13

a 13

Example (cont)

Use the lengths of the three sides to find the other five ratios.

sin 6

7

cos 13

7

tan 6

13

6 13

13

csc 7

6

sec 7

13

7 13

13

cot 13

6

Function Values of 45º

A right triangle with one 45º, must have a second 45º, making it an isosceles triangle, with legs the same length. Consider one with legs of length 1.

sin 45ºopp

hyp

1

2

2

20.7071

45º

45º

1

1

2 cos 45ºadj

hyp

1

2

2

20.7071

tan 45ºopp

adj

1

11

Function Values of 30º

A right triangle with 30º and 60º acute angles is half an equilateral triangle. Consider an equilateral triangle with sides 2 and take half of it.

sin 30º1

20.5,

30º

60º

2

1

3cos 30º

3

20.8660,

tan 30º1

3

3

30.5774

Function Values of 60º

A right triangle with 30º and 60º acute angles is half an equilateral triangle. Consider an equilateral triangle with sides 2 and take half of it.

sin60º3

20.8660,

cos60º1

20.5,

3tan 60º 3 1.7321

1

30º

60º

2

1

3

Example

As a hot-air balloon began to rise, the ground crew drove 1.2 mi to an observation station. The initial observation from the station estimated the angle between the ground and the line of sight to the balloon to be 30º. Approximately how high was the balloon at that point? (We are assuming that the wind velocity was low and that the balloon rose vertically for the first few minutes.)

Solution:

Draw the situation, label the acute angle and length of the adjacent side.

Example (cont)

tan 30ºopp

adjh

1.2

1.2 tan 30ºh

1.23

3h

0.7 h

The balloon is approximately 0.7 mi, or 3696 ft, high.

Function Values of Any Acute Angle

Angles are measured either in degrees, minutes, and seconds: 1º = 60´, 1´ = 60´´; referred to as the DºM´S´´ form

61 degrees, 27 minutes, 42 seconds 61º 27 42

or are measured in decimal degree form, expressing the fraction parts of degrees in decimal form

61º 27 42 61.451

Examples

Find the trigonometric function value, rounded to four decimal places, of each of the following:

0.5703899297

Solution:

Check that the calculator is in degree mode.

a) tan29.7º

a) tan29.7º 0.5704

b) sec 48º c) sin84º1 0 3 9

1

cos 48ºb) sec 48º 1.49447655

c) sin84º1 0 3 9

1.49445

0.9948409474 0.9948

Example

A window-washing crew has purchased new 30-ft extension ladders. The manufacturer states that the safest placement on a wall is to extend the ladder to 25 ft and to position the base 6.5 ft from the wall. What angle does the ladder make with the ground in this position?

Solution:

Draw the situation, label the hypotenuse and length of the side adjacent to .

Example (cont)

cos adj

hyp

6.5 ft

25 ft

0.26

74.92993786º

Thus when the ladder is in its safest position, it makes an angle of about 75º with the ground.

Use a calculator to find the acute angle whose cosine is 0.26:

Cofunction Identities

Two angles are complementary whenever the sum of their measures is 90º. Here are some relationships.

sin cos 90º

90º – cos sin 90º tan cot 90º cot tan 90º sec csc 90º csc sec 90º

Example

Given that sin 18º ≈ 0.3090, cos 18º ≈ 0.9511, and tan 18º ≈ 0.3249, find the six trigonometric function values of 72º.

csc18º1

sin18º3.2361

Solution:

sec18º1

cos18º1.0515

cot18º1

tan18º3.0777

sin 72ºcos18º0.9511

cos 72ºsin18º0.3090

tan 72ºcot18º3.0777

cot 72ºtan18º0.3249

sec 72ºcsc18º3.2361

csc 72ºsec18º1.0515