lat04_0502.ppt

Copyright © 2009 Pearson Addison-Wesley 5.2-1

5Trigonometric Functions


5.1 Angles

5.2 Trigonometric Functions

5.3 Evaluating Trigonometric Functions

5.4 Solving Right Triangles

5 Trigonometric Functions

Copyright © 2009 Pearson Addison-Wesley

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Trigonometric Functions5.2Trigonometric Functions ▪ Quadrantal Angles ▪ Reciprocal Identities ▪ Signs and Ranges of Function Values ▪ Pythagorean Identities ▪ Quotient Identities


Trigonometric Functions

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


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Trigonometric Functions

The six trigonometric functions of θ are defined as follows:


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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 FINDING FUNCTION VALUES OF AN ANGLE


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Example 1 FINDING FUNCTION VALUES OF AN ANGLE (continued)


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



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Use the definitions of the trigonometric functions.


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

Choose x = 2.

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.


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The point (2, –1) lies on the terminal side, and the

corresponding value of r is


Multiply by to rationalize

the denominators.


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Example 4(a) FINDING FUNCTION VALUES OF QUADRANTAL ANGLES

The terminal side passes through (0, 1). So x = 0, y = 1, and r = 1.

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


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Example 4(b) FINDING FUNCTION VALUES OF QUADRANTAL ANGLES

x = –3, y = 0, and r = 3.

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


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Undefined Function Values

If the terminal side of a quadrantal angle lies along the y-axis, then the tangent and secant functions are undefined.

If the terminal side of a quadrantal angle lies along the x-axis, then the cotangent and cosecant functions are undefined.


Commonly Used Function Values

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csc sec cot tan cos sin


Using a Calculator

A calculator in degree mode returns the correct values for sin 90° and cos 90°.

The second screen shows an ERROR message for tan 90° because 90° is not in the domain of the tangent function.


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Caution

One of the most common errors involving calculators in trigonometry occurs when the calculator is set for radian measure, rather than degree measure.


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Reciprocal Identities

For all angles θ for which both functions are defined,


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Example 5(a) USING THE RECIPROCAL IDENTITIES

Since cos θ is the reciprocal of sec θ,


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Example 5(b) USING THE RECIPROCAL IDENTITIES

Since sin θ is the reciprocal of csc θ,

Rationalize the denominator.


Signs of Function Values

++IV

++III

++II

++++++I

csc sec cot tan cos sin in

Quadrant


Signs of Function Values


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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.

Since sin > 0 in quadrants I and II, and tan < 0 in quadrants II and IV, both conditions are met only in quadrant II.

(b) cos < 0, sec < 0

The cosine and secant functions are both negative in quadrants II and III, so could be in either of these two quadrants.


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Ranges of Trigonometric Functions


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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

Impossible

Possible

(c) sec θ = .6 Impossible


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Pythagorean Identities

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


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Quotient Identities

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


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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.

Since θ is in quadrant III, both sin θ and cos θ are negative.


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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)


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Use the identity to find sec θ. Then use the reciprocal identity to find cos θ.

Choose the negative square root since sec θ <0 for θ in quadrant III.

Secant and cosine are reciprocals.


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Choose the negative square root since sin θ <0 for θ in quadrant III.


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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.

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lat04_0502.ppt