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

Holt McDougal Geometry

8-2-EXT Trigonometric Ratios and Complementary Angles8-2-EXTTrigonometric Ratios and Complementary Angles

Holt Geometry

Lesson PresentationLesson Presentation



8-2-EXT Trigonometric Ratios and Complementary Angles

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

Objectives



cofunction

Vocabulary



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°.



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

hypotenuse1237

=



Example 1: Continue

Then, find the sine and cosine of ∠B.

cos A =adjacent

hypotenuse3537

=

sin B =opposite

hypotenuse3537

=

cos B =adjacent

hypotenuse1237

=



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

hypotenuse1026

=5

13=

1213

=cos A =adjacent

hypotenuse2426

=



Check It Out! Example 1 Continued

513

=Cos B =adjacent

hypotenuse1026

=

sin B =opposite

hypotenuse2426

=1213

=



The trigonometric function of the complement of an angle is called a cofunction. The sine and cosines are cofunctions of each other.



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




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



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 + 5 + 4x + 10 = 90 5x + 15 = 90 5x = 75 x = 15



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°.




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 + 2 + x + 44 = 90 4x + 46 = 90

4x = 44 x = 11






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°.




2x + 20 + 3x + 30 = 90 5x + 50 = 90 5x = 40 x = 8



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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