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14.3 Trigonometric Functions

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Objectives Find the values of the 6 trigonometric functions of an angle Find the trigonometric function values of a quadrantal angle

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The 6 Trigonometric FUNctions To define our 6 trig functions we will use the symbol θ (theta) to represent an angle in standard position and any point P(x,y) on the terminal side of our angle. A perpendicular from P to the x-axis at point Q determines a right triangle using the origin O as the third angle. To find the length of the hypotenuse (r) we can use the distance formula. The six trig functions are sine, cosine, tangent, cosecant, secant, and cotangent

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6 Trig FUNctions cont… Let (x,y) be a point other than the origin on the terminal side of an angle θ in standard position. The six trigonometric functions are defined as follows…

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Example 1 The terminal side of an angle θ in standard position passes through the point (12,5). Find the values of the six trigonometric functions of angle θ. Draw it!

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Example 2 The terminal side of an angle θ in standard position passes through the point (-3,-4) Find the values of the six trigonometric functions of angle θ.

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Example 3 (Give it a go!) The terminal side of an angle θ in standard position passes through the point (8,-6) Find the values of the six trigonometric functions of angle θ.

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Using the Equation of a line to find Trig Functions Remember from Algebra 1 that the graph of the line ax+by=0 passes through the origin. Because we are dealing with angles, we want to work with rays not lines. To do this we will restrict our x values to be x≥0 or x≤0 By choosing a point on the ray we can find the trig funtions.

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Example 4 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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Example 5 (Give it a go!) Find the six trigonometric function values of the angle θ in standard position if the terminal side of θ is defined by 3x-2y=0, x≤0.

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Slope and Tangent Change the equation of the line from example 4 to slope intercept form. x + 2y = 0 What is the slope of this line? What do you notice? In general, it is true that tan θ = m Note: The values we found in our first three examples are exact. If we use the calculator to approximate, our decimal answers would not be acceptable if we were looking for exact values.

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Determining Trig values of Quadrantal Angles Looking at the quadrantal angles you can see that some of the x or y values will be 0. This will cause some of the trig functions to be undefined. We have already shown that we can use any point on the terminal side to find the function, so we choose to use 1 wherever possible.

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Example 6a Find the value of the six trig functions for an angle of 90°

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Example 6b Find the value of the six trig 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 it lies along the x-axis, then the cotangent and the cosecant functions are undefined. You can find a table on page 680 of your book that has the most commonly used quadrantal angles (0°,90°,180°, 270°, and 360°) and their function values.

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CW/HW Page 681 #4, 6, 13, 46, 74

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