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Lesson 8-6 The Sine and Cosine Ratios (page 312) The sine ratio and cosine ratio relate the legs to the hypotenuse . How can trigonometric ratios be used to find sides and angles of a triangle? The tangent ratio is the ratio of the lengths of the legs . B c a Review! A b leg C leg In relationship to angle A … c A B a b adjacent leg C opposite leg opposite leg vs adjacent leg In relationship to angle B … c A B a b opposite leg C adjacent leg opposite leg vs adjacent leg Definition of Tangent Ratio B c a Review! A b tangent of ∠A = tan A C a BC = = AC b length of the leg opposite ÐA tan A = length of the leg adjacent to ÐA Definition of Sine Ratio B c A a b sine of ∠A = sin A C a BC = = AB c length of the leg opposite ÐA sin A = length of the hypotenuse Definition of Cosine Ratio B c A a b cosine of ∠A = cos A C b AC = = AB c length of the leg adjacent to ÐA cos A = length of the hypotenuse B c a A C b length of the leg opposite ÐA sin A = length of the hypotenuse length of the leg adjacent to ÐA cos A = length of the hypotenuse length of the leg opposite ÐA tan A = length of the leg adjacent to ÐA B REMEMBER! A opposite sin = hypotenuse c a b adjacent cos = hypotenuse DO NOT FORGET! C opposite tan = adjacent SOH CAH TOA Example 1 Express the sine and cosine of ∠A & ∠B as ratios. B O 5 ____ C 5 13 (a) sin A = ______ (c) sin B = ______ BC + 12 = 13 2 BC +144 =169 2 13 H 2 BC = 25 2 BC = 5 12 A 2 A 12 13 (b) cos A = ______ (d) cos B = ______ Example 1 Express the sine and cosine of ∠A & ∠B as ratios. B A 5 ____ C 5 13 (a) sin A = ______ 12 13 (c) sin B = ______ BC +12 =13 2 13 H 2 BC +144 =169 2 BC = 25 2 BC = 5 12 O 2 A 12 13 (b) cos A = ______ 5 13 (d) cos B = ______ Example 2 Use the trigonometry table, then use a calculator to find the approximate decimal values. “≈” means “is approximately equal to” 0.3746 (a) sin 22º ≈ __________ 0.1908 (b) cos 79º ≈ __________ Please note that these are only APPROXIMATIONS! Now try this with a calculator! To enter this in your calculator you will need to use the SIN or COS function key. Enter SIN(22) then press ENTER (=) and round to 4 decimal places. (a) 0.3746 sin 22º ≈ ____________ Enter COS(79) then press ENTER (=) and round to 4 decimal places. (b) 0.1908 cos 79º ≈ ____________ … Example 2 Use the trigonometry table to find the approximate angle measures. “≈” means “is approximately equal to” 61º ≈ 0.8746 (c) sin ______ 39º ≈ 0.7771 (d) cos _______ Please note that these are only APPROXIMATIONS! Now try this with a calculator! To enter this in your calculator you will need to use the inverse key or 2nd function key. Enter SIN-1(.8746) then press ENTER (=) and round to the nearest degree 61º ≈ 0.8746 (c) sin ______ Enter COS-1(.7771) then press ENTER (=) and round to the nearest degree 39º ≈ 0.7771 (d) cos _______ What can you say about the values for the sine or cosine of an angle? The values for sine and cosine … Think about it … if the hypotenuse is the longest side and it is the denominator of the ratios, then it … … will always be less than 1. Enter SIN-1(1.5) then press ENTER (=) … Example 3 (a) Find the value of x and y to the nearest integer. 52 x ≈ ______ y ≈ ______ x sin 38º = 84 84 38º x x = 84 × sin38º x » 51.7 y Example 3 (a) Find the value of x and y to the nearest integer. 52 x ≈ ______ 66 y ≈ ______ y cos 38º = 84 84 38º x y y = 84 × cos38º y » 66.19 Example 3 (b) Find the value of x and y to the nearest integer. 12 x ≈ ______ y ≈ ______ 7 cos 55º = 1 x x 55º x y 7 14 7 7 x= 1 cos55º x »12.2 Example 3 (b) Find the value of x and y to the nearest integer. 12 x ≈ ______ 10 y ≈ ______ y tan 55º = 7 x 55º x y 7 14 7 y = 7× tan55º y » 9.997 Example 4 (a): Find “n” to the nearest degree. 44 n ≈ ______ 14 sin nº = 20 nº H 20 14 O æ 14 ö n = sin ç ÷ è 20 ø -1 n » 44.4 Example 4 (b) Find the measures of the 3 angles of a 3-4-5 ∆. Example 4 (b) Find the measures of the 3 angles of a 3-4-5 ∆. 5 ? 3 +4 2 5 3 4 2 2 25 ? 9 + 16 25 = 25 ∴ a right triangle. continue Example 4 (b) Find the measures of the 3 angles of a 3-4-5 ∆. nº 5 3 4 sin nº = 5 æ 4ö n = sin ç ÷ è 5ø -1 4 90º - 53º = 37º n » 53 ∴ the angle measures are 37º, 53º, & 90º. OPTIONAL Assignment Written Exercises on pages 314 to 316 1 to 17 odd numbers ~ #20 is BONUS! ~ Trigonometry Worksheet #1 Lessons 8-5 & 8-6 The Sine, Cosine, and Tangent Ratios How can trigonometric ratios be used to find sides and angles of a triangle?