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# Download 8.2A Trigonometric Ratios

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8.2A Trigonometric Ratios Objectives: G.SRT.6: Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. For the Board: You will be able to find the sine, the cosine, and the tangent of an acute angle. Anticipatory Set: By AA Similarity, a right triangle with a given acute angle, such as 32°, is similar to every other right triangle with that same acute angle measure. L X A ΔABC ~ ΔLMN ~ ΔXYZ M 32° C B 32° N Y Since the triangles are similar . . . 32° The ratio of the short leg to the hypotenuse in each of these triangles will be the same AC LN XZ . AB LM XY The ratio of the long leg to the hypotenuse in each of these triangles will be the same BC MN YZ . AB LM XY The ratio of the short leg to the long leg in each of these triangles will be the same AC LN XZ . BC MN YZ These ratios are called trigonometric ratios. A trigonometric ratio is a ratio of the lengths of two sides of a right triangle. The three basic trigonometric ratios are sine, cosine, and tangent, which are abbreviated as sin, cos, and tan. Trigonometric Ratios Let ABC be a right triangle. The sine, the cosine, and the tangent of the acute angles are defined as follows. B c a C b A Z The sine of an angle is the ratio of the length of the leg opposite the angle to the length of the hypotenuse. sin A Opposite Leg a Hypotenuse c sin B Opposite Leg b Hypotenuse c The cosine of an angle is the ratio of the length of the leg adjacent to the angle to the length of the hypotenuse. cos A Adjacent Leg b Hypotenuse c cos B Adjacent Leg a Hypotenuse c The tangent of an angle is the ratio of the length of the leg opposite the angle to the length of the leg adjacent to the angle. tan A Opposite Leg a AdjacentLeg b tan B Opposite Leg b AdjacentLeg a An often used method of remembering these is the word SOH-CAH-TOA. Open the book to page 541 and read example 1. Example: Write each trigonometric ratio as a fraction and as a decimal rounded to the nearest hundredth. J a. sin J 60/61 0.98 61 11 b. cos J 11/61 0.18 c. tan K 11/60 0.18 L 60 White Board Activity: Additional Practice: Write each trigonometric ratio as a fraction and as a decimal rounded to the nearest hundredth. X a. sin Y 5/13 0.38 b. cos X 5/13 0.38 13 5 c. tan Y 5/12 0.42 Y Assessment: Question Student pairs. Independent Practice: Text: pgs. 545 – 548 prob. For a Grade: Text: pgs. 545 – 548 prob. 12 Z K