copyright © 2009 pearson addison-wesley 1.3-1 1 trigonometric functions
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Copyright © 2009 Pearson Addison-Wesley 1.3-1
1Trigonometric Functions
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1.1 Angles
1.2 Angle Relationships and Similar Triangles
1.3 Trigonometric Functions
1.4 Using the Definitions of the Trigonometric Functions
1 Trigonometric Functions
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1.3-3
Trigonometric Functions1.3Trigonometric Functions ▪ Quadrantal Angles
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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 .
Example 2 FINDING FUNCTION VALUES OF AN ANGLE
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Example 2 FINDING FUNCTION VALUES OF AN ANGLE (continued)
Use the definitions of the trigonometric functions.
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Example 3 FINDING FUNCTION VALUES OF AN ANGLE
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
Example 3 FINDING FUNCTION VALUES OF AN ANGLE (continued)
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.
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Commonly Used Function Values
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1undefined0undefined01270
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1undefined0undefined0190
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csc sec cot tan cos sin
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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.