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Holt McDougal Geometry 8-2-EXT Trigonometric Ratios and Complementary Angles 8-2-EXT Trigonometric Ratios and Complementary Angles Holt Geometry Lesson Presentation Lesson Presentation Holt McDougal Geometry

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8-2-EXT Trigonometric Ratios and Complementary Angles Use the relationship between the sine and cosine of complementary angles. Objectives

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Holt McDougal Geometry 8-2-EXT Trigonometric Ratios and Complementary Angles cofunction Vocabulary

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Holt McDougal Geometry 8-2-EXT Trigonometric Ratios and Complementary Angles The acute angles of a right triangle are complementary angles. If the measure of one of the two acute angles is given, the measure of the second acute angle can be found by subtracting the given measure from 90°.

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Holt McDougal Geometry 8-2-EXT Trigonometric Ratios and Complementary Angles Example 1: Finding the Sine and Cosine of Acute Angles Find the sine and cosine of the acute angles in the right triangle shown. Start with the sine and cosine of ∠ A. sin A = opposite hypotenuse =

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Holt McDougal Geometry 8-2-EXT Trigonometric Ratios and Complementary Angles Example 1: Continue Then, find the sine and cosine of ∠ B. cos A = adjacent hypotenuse = sin B = opposite hypotenuse = cos B = adjacent hypotenuse =

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Holt McDougal Geometry 8-2-EXT Trigonometric Ratios and Complementary Angles Check It Out! Example 1 Find the sine and cosine of the acute angles of a right triangle with sides 10, 24, 26. (Use A for the angle opposite the side with length 10 and B for the angle opposite the side with length 24.) sin A = opposite hypotenuse = 5 13 = = cos A = adjacent hypotenuse =

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Holt McDougal Geometry 8-2-EXT Trigonometric Ratios and Complementary Angles Check It Out! Example 1 Continued 5 13 = Cos B = adjacent hypotenuse = sin B = opposite hypotenuse = =

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Holt McDougal Geometry 8-2-EXT Trigonometric Ratios and Complementary Angles The trigonometric function of the complement of an angle is called a cofunction. The sine and cosines are cofunctions of each other.

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Holt McDougal Geometry 8-2-EXT Trigonometric Ratios and Complementary Angles Example 2: Writing Sine in Cosine Terms and Cosine in Sine Terms A. Write sin 52° in terms of the cosine. sin 52° = cos(90 – 52)° = cos 38 B. Write cos 71° in terms of the sine. cos 71° = sin(90 – 71)° = sin 19

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Holt McDougal Geometry 8-2-EXT Trigonometric Ratios and Complementary Angles Check It Out! Example 2 A. Write sin 28° in terms of the cosine. sin 28° = cos(90 – 28)° = cos 62 B. Write cos 51° in terms of the sine. cos 51° = sin(90 – 51)° = sin 39

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Holt McDougal Geometry 8-2-EXT Trigonometric Ratios and Complementary Angles Example 3: Finding Unknown Angles Find the two angles that satisfy the equation below. sin (x + 5)° = cos (4x + 10)° If sin (x + 5)° = cos (4x + 10)° then (x + 5)° and (4x + 10)° are the measures of complementary angles. The sum of the measures must be 90°. x x + 10 = 90 5x + 15 = 90 5x = 75 x = 15

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Holt McDougal Geometry 8-2-EXT Trigonometric Ratios and Complementary Angles Example 3: Continued Substitute the value of x into the original expression to find the angle measures. The measurements of the two angles are 20° and 70°.

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Holt McDougal Geometry 8-2-EXT Trigonometric Ratios and Complementary Angles Check It Out! Example 3 Find the two angles that satisfy the equation below. A. sin(3x + 2)° = cos(x + 44)° If sin(3x + 2)° = cos(x + 44)° then (3x + 2)° and (x + 44)° are the measures of complementary angles. The sum of the measures must be 90°. 3x x + 44 = 90 4x + 46 = 90 4x = 44 x = 11

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Holt McDougal Geometry 8-2-EXT Trigonometric Ratios and Complementary Angles Check It Out! Example 3 Continued Substitute the value of x into the original expression to find the angle measures. The measurements of the two angles are 35° and 55°. B. sin(2x + 20)° = cos(3x + 30)° If sin(2x + 20)° = cos(3x + 30)° then (2x + 20)° and (3x + 30)° are the measures of complementary angles. The sum of the measures must be 90°.

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Holt McDougal Geometry 8-2-EXT Trigonometric Ratios and Complementary Angles Check It Out! Example 3 Continued 2x x + 30 = 90 5x + 50 = 90 5x = 40 x = 8 Substitute the value of x into the original expression to find the angle measures. The measurements of the two angles are 36° and 54°.

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