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**Trigonometric Ratios and Complementary Angles**

8-2-EXT Lesson Presentation Holt McDougal Geometry Holt Geometry

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Objectives Use the relationship between the sine and cosine of complementary angles.

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Vocabulary cofunction

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**The acute angles of a right triangle are 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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**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 12 37 =

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Example 1: Continue cos A = adjacent hypotenuse 35 37 = Then, find the sine and cosine of ∠B. sin B = opposite hypotenuse 35 37 = cos B = adjacent hypotenuse 12 37 =

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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 10 26 = 5 13 12 13 = cos A = adjacent hypotenuse 24 26

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**Check It Out! Example 1 Continued**

sin B = opposite hypotenuse 24 26 = 12 13 5 13 = Cos B = adjacent hypotenuse 10 26

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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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**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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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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**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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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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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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**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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**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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