Evaluating trigonometric functions

MATH 1110

February 6, 2009

Evaluating trigonometric functions

Remark. Throughout this document, remember the angle measurement convention, which states that if the measurement of an angle appears without units, then it

is assumed to be measured in radians.

Contents

1 Acute and square angles

1

2 Larger angles ¡ª the geometric method

2

3 Larger angles ¡ª the formulas method

5

1

Acute and square angles

You will need to memorize the values of sine and cosine at the following acute angles:

¦Ð

¦Ð

¦Ð

= 30? , = 45? , and = 60? . You will also need to know their values at the square

6

4

3

3¦Ð

?

? ¦Ð

angles 0 = 0 , = 90 , ¦Ð = 180? ,

= 270? , and 2¦Ð = 360? .

2

2

¦È

sin ¦È

0

¦Ð

6

¦Ð

4

0

1

2

1

¡Ì

2

¡Ì

cos ¦È

1

3

2

1

¡Ì

2

¦Ð

3

¦Ð

2

¦Ð

3¦Ð

2

2¦Ð

3

2

1

0

?1

0

1

2

0

?1

0

1

¡Ì

The sine and cosine of the square angles are easy to figure out from the definition of

cosine and sine as the x and y coordinates of points on the unit circle. (See Figure P.66

on page 45 of the textbook.) The sine and cosine of the acute angles listed above can

be found by studying a 30? -60? -90? triangle and a 45? -45? -90? triangle. (See Examples

3 and 4 on page 47 of the textbook.)

1

MATH 1110

February 6, 2009

Values of the other trigonometric functions at the angles listed above can be found

easily, since the other functions are all built from sine and cosine.

Example 1.

Question.

¦Ð

¦Ð

Evaluate tan and sec .

3

4

Answer.

sin ¦È

Since tan ¦È =

, we have

cos ¦È

¡Ì

sin ¦Ð3

¦Ð

3/2 ¡Ì

tan =

=

= 3.

3

cos ¦Ð3

1/2

Similarly,

sec

2

¡Ì

¦Ð

1

1

¡Ì = 2.

=

¦Ð =

4

cos 4

1/ 2

Larger angles ¡ª the geometric method

The first thing to notice is that since sine and cosine repeat their values every 2¦Ð

radians, if you are asked to evaluate one of these functions at an angle which is not

between 0 and 2¦Ð, you can simply add or subtract 2¦Ð until the angle does lie in that

interval. For example,













 

10¦Ð

10¦Ð 6¦Ð

10¦Ð

4¦Ð

sin

? 2¦Ð = sin

?

= sin

= sin

.

3

3

3

3

3

The next thing to notice is that the value of sine or cosine at any angle ¦È with

0 ¡Ü ¦È ¡Ü 2¦Ð is essentially determined by their values at an a particular acute angle

related to ¦È, called the reference angle of ¦È.

Definition 2.1.

The reference angle of a non-square angle ¦È is the acute angle formed between the

x-axis and the ray from the origin making an angle of ¦È with the positive x-axis ray.

Recall that the xy-plane is traditionally divided into four quadrants, hence dividing

the non-square angles in one rotation into four types.

2

MATH 1110

February 6, 2009

Quadrant

angles ¦È included

¦Ð

2

reference angle of ¦È

I

0 ................
................

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