8 2 Trigonometric Ratios Warm Up Lesson Presentation

8 -2 Trigonometric. Ratios Warm Up Lesson Presentation Lesson Quiz Holt. Mc. Dougal Geometry Holt

8 -2 Trigonometric Ratios Warm Up Write each fraction as a decimal rounded to the nearest hundredth. 1. 2. 0. 67 0. 29 Solve each equation. 3. x = 7. 25 Holt Mc. Dougal Geometry 4. x = 7. 99

8 -2 Trigonometric Ratios Objectives Find the sine, cosine, and tangent of an acute angle. Use trigonometric ratios to find side lengths in right triangles and to solve real-world problems. Use the relationship between the sine and cosine of complementary angles. Holt Mc. Dougal Geometry

8 -2 Trigonometric Ratios Vocabulary trigonometric ratio sine cosine tangent cofunction Holt Mc. Dougal Geometry

8 -2 Trigonometric Ratios By the AA Similarity Postulate, a right triangle with a given acute angle is similar to every other right triangle with that same acute angle measure. So ∆ABC ~ ∆DEF ~ ∆XYZ, and. These are trigonometric ratios. A trigonometric ratio is a ratio of two sides of a right triangle. Holt Mc. Dougal Geometry

8 -2 Trigonometric Ratios Holt Mc. Dougal Geometry

8 -2 Trigonometric Ratios Writing Math In trigonometry, the letter of the vertex of the angle is often used to represent the measure of that angle. For example, the sine of A is written as sin A. Holt Mc. Dougal Geometry

8 -2 Trigonometric Ratios Example 1 A: Finding Trigonometric Ratios Write the trigonometric ratio as a fraction and as a decimal rounded to the nearest hundredth. sin J Holt Mc. Dougal Geometry

8 -2 Trigonometric Ratios The hypotenuse is always the longest side of a right triangle. So the denominator of a sine or cosine ratio is always greater than the numerator. Therefore the sine and cosine of an acute angle are always positive numbers less than 1. Since the tangent of an acute angle is the ratio of the lengths of the legs, it can have any value greater than 0. Holt Mc. Dougal Geometry

8 -2 Trigonometric Ratios 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°. Holt Mc. Dougal Geometry

8 -2 Trigonometric Ratios 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. opposite sin A = hypotenuse Holt Mc. Dougal Geometry 12 = 37

8 -2 Trigonometric Ratios Example 1: Continue adjacent cos A = hypotenuse = 35 37 Then, find the sine and cosine of ∠B. opposite sin B = hypotenuse adjacent cos B = hypotenuse Holt Mc. Dougal Geometry 35 = 37 = 12 37

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

8 -2 Trigonometric Ratios 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 Holt Mc. Dougal Geometry

8 -2 Trigonometric Ratios 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 Holt Mc. Dougal Geometry

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

8 -2 Trigonometric Ratios 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°. Holt Mc. Dougal Geometry