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Aim #24: What is the relationship between the sine and cosine CC Geometry H of complementary angles? Do Now: Given the diagram of the right triangle, complete the following table. Express all ratios in simplest radical form. Angle Measure sin θ cos θ tan θ s 2 s 4 t t a) Which values are equal? b) How are tan t and tan s related? How are the two co-functions (sine and cosine) related? In right triangle ABC, the measurement of acute angle ≮A is denoted by α (alpha), and the measurement of acute angle ≮B is denoted by β (beta). B α and β are a pair of _____________________ angles Determine the following ratios: c A a b C • Since the ratios for sine α & cosine β are the same, ________ = ________. • Since the ratios for cosine α & sine β are the same, ________ = ________ . The sine of an angle is equal to the ____________ of its ________________, and the______________ of an angle is equal to the ______ of its complement. ∴For complementary angles α and β, sin α = cos ____ and cos ___ = sin β. Given measure θ such that 0 < θ < 90, cos (θ) = sin (90 - θ) and sin θ = cos (90 - θ). The two complementary angles in a right triangle are the ________ angles. The co- prefix in cosine refers to the fact that the cosine of an angle equals the sine of its ___________________. Consider right triangle ABC with right ≮C, and the degree measures of ≮A and ≮B are α and β, respectively. a) Find α + β. _____ b) Use trig. ratios to express c) Use trig. ratios to express two different ways. 5 4 two different ways. 3 d) What can you conclude about sin α and cos β ? e) What can you conclude about cos α and sin β ? 2. Find value of θ that make each statement true. a) sin θ = cos (25) b) sin 80 = cos θ c) sin θ = cos (θ + 10) d) cos(θ - 45) = sin θ e) cos θ = sin (θ - 30) f) sin (θ + 20) = sin ( θ + 40) 3. For what angle measurement must sine and cosine have the same value. Explain how you know. 0 0 What are sine and cosine values for θ = 0 and 90 ? In the figure to the right, the hypotenuse c of right ΔABC is the radius of a circle which has a radius of length 1 unit. 1 • c = ___ • sin θ = = and cos θ = = . 0 As θ gets closer to 0 , a __________ and sin θ approaches ___. 0 As θ gets closer to 0 , b __________ and cos θ approaches ___. 0 0 Definitions: sin 0 = _____ and cos 0 = _____ 0 As θ gets closer to 90 , a __________ and gets closer to ___, and the value of sin θ approaches ___. 0 As θ gets closer to 90 , b __________ and gets closer to ___, and the value of cos θ approaches ___. 0 0 Definitions: sin 90 = _____ and cos 90 = _____ 0 0 0 0 **Since sin 0 = cos 90 , and cos 0 = sin 90 , this is consistent with the fact that the sine of an angle equals the_____________________________________.** What are the exact sine and cosine values for the "special angles"? 0 0 a) Write in the sine and cosine value for 0 and 90 : 300 00 θ 600 450 900 sin θ cos θ 0 0 0 b) Determine the exact sine and cosine value for 30 , 60 , and 45 using the equilateral triangle with side 2 and isosceles right triangle with side 1 below. Write your answers, with rationalized denominators, in the chart above. 450 300 300 2 1 600 600 1 **Memorize the values of sine and cosine for the special angles above.** To help memorize, note that the complementary angles have the same values, but in reverse order. • sin 0 = cos 90 and sin 90 = cos 0 • sin 30 = cos 60 and sin 60 = cos 30 • sin 45 = cos 45 0 0 0 0 0 0 0 0 0 300 00 θ 600 450 900 sin θ cos θ 1. The triangles below are special right triangles. Find the unknown lengths a and b, using sin and cos values of an acute angle. Show your solving of equations. a) b) 3 a c 3 300 b 600 a c) e) d) c a b 45 a 45 0 450 c 0 a Let's Sum it Up! • The sine of an angle is equal to the cosine of its complementary angle, and the cosine of and angle is equal to the sine of its complementary angle. • Sin 90 = 1 and cos 0 = 1 and similarly, sin 0 = 0 and cosine of 90 = 1. 0 0 0 0 • The values for the cosine and sine values for the special angles are the same, but they are in reverse order. Name_____________________ Date _____________________ CC Geometry H HW #24 #1-6 Find the values of θ that make the equation true. 1. sin θ = cos 32 2. cos 11 = sin θ 3. sin θ = cos (θ + 38) 3. sin (θ + 10) = sin 60 4. cos θ = sin (3θ + 20) 6. #7-12 The triangles below are special right triangles. Find the unknown lengths using sin and cos values of an acute angle. Show your solving of an appropriate equation. 7. 8. 600 9. 300 60 0 y 7 y x x x y 12 11. 10. 12. 10 x 450 45 x y 0 x 45 0 y y OVER 13. A square has side lengths . Use sine or cosine to find the length of the diagonal of the square. Confirm your answer using the relationships we know in 45-45-90 right triangles. 14. Use the sine and cosine values of special angles to find the exact sum of: a) sin 30 + cos 60 b) sin 60 + cos 30 15. Find the value of R that will make the equation sin 63° = cos R true when 0° < R < 90°. Explain your answer. Mixed Review: 1) Given an equilateral triangle with sides of length 9, find the length of the altitude. 2) Which statement describe the properties of a rhombus? Select all that apply. a. The diagonals bisect the angles. b. The diagonals bisect each other. c. The opposite sides are parallel. d. The opposite angles are congruent. e. The diagonals are congruent. f. The diagonals are perpendicular.