5.2-1

5.2Trigonometric Functions

Let (x, y) be a point other then the origin on the terminal side of an angle in standard position. The distance from the point to the origin is

5.2-2

The six trigonometric functions of θ are defined as follows:

5.2-3

The terminal side of angle in standard position passes through the point (8, 15). Find the values of the six trigonometric functions of angle .

Example 1

5.2-4

The terminal side of angle in standard position passes through the point (–3, –4). Find the values of the six trigonometric functions of angle .

Example 2

5.2-5

Example 3

We can use any point on the terminal side of to find the trigonometric function values.

Find the six trigonometric function values of the angle θ in standard position, if the terminal side of θ is defined by x + 2y = 0, x ≥ 0.

5.2-6

Example 4

If the terminal side of angle θ in standard position lies on the line defined by 3x + 4y = 0, x ≥ 0. Then the value of 5sin + 10cos is

A)5

B)9.5

C)11

D)-4

E)-3

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Example 5(a) FINDING FUNCTION VALUES OF QUADRANTAL ANGLES

Find the values of the six trigonometric functions for an angle of 90°.

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Example 5(b) FINDING FUNCTION VALUES OF QUADRANTAL ANGLES

Find the values of the six trigonometric functions for an angle θ in standard position with terminal side through (–3, 0).

5.2-9

Commonly Used Function Values

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1undefined0undefined01270

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csc sec cot tan cos sin

5.2-10

Reciprocal Identities

For all angles θ for which both functions are defined,

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Example 5 USING THE RECIPROCAL IDENTITIES

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Signs of Function Values

++IV

++III

++II

++++++I

csc sec cot tan cos sin in

Quadrant

5.2-13

Signs of Function Values

5.2-14

Identify the quadrant (or quadrants) of any angle that satisfies the given conditions.

Example 6 IDENTIFYING THE QUADRANT OF AN ANGLE

(a) sin > 0, tan < 0.

(b) cos < 0, sec < 0

5.2-15

Ranges of Trigonometric Functions

5.2-16

Decide whether each statement is possible or impossible.

Example 7 DECIDING WHETHER A VALUE IS IN THE RANGE OF A TRIGONOMETRIC FUNCTION

(a) sin θ = 2.5

(b) tan θ = 110.47

(c) sec θ = 0.6

(c) sec θ = 0.6

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Pythagorean Identities

For all angles θ for which the function values are defined,

5.2-18

Quotient Identities

For all angles θ for which the denominators are not zero,

5.2-19

Example 8 FINDING OTHER FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT

Find sin θ and cos θ, given that and θ is in quadrant III.

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Caution

It is incorrect to say that sin θ = –4 and cos θ = –3, since both sin θ and cos θ must be in the interval [–1, 1].

Example 8 FINDING OTHER FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT (continued)

5.2-21

Example 8 FINDING OTHER FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT (continued)

This example can also be worked by drawing θ in standard position in quadrant III, finding r to be 5, and then using the definitions of sin θ and cos θ in terms of x, y, and r.