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Example 4: The following diagrams show the basic side lengths of the 30-60-90 and 45-45-90 triangles. 600 450 2 1 1 300 450 1 Determine side lengths of three different triangles similar to each of the triangle in the chart. Then generalize the side lengths in terms of x. 306090 Triangle side lengths 1: : 2 454590 Triangle side lengths 1: 1 : ______ : ______ : ______ ______ : ______ : ______ ______ : ______ : ______ ______ : ______ : ______ ______ : ______ : ______ ______ : ______ : ______ x : ______ : ______ x : ______ : ______ Exercises 1. The triangles below are 30-60-90 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) a b 45 c 450 a c a 0 450 2. Given an equilateral triangle with sides of length 9, find the length of the altitude. Confirm your answer using the Pythagorean Theorem. 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 900 = 1 and cos 00 = 1 and similarly, sin 00 = 0 and cosine of 900 = 1. • 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 #27 #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 30-60-90 right triangles. Find the unknown lengths x and y, using sin and cos values of an acute angle. Show your solving of an appropriate equation. 9. 8. 7. y 300 600 600 y 7 x x x y 12 11. 10. 12. 10 x 450 y 450 x x 45 0 y y OVER 13. A square has side lengths 7√2. Use sine or cosine to find the length of the diagonal of the square. Confirm your answer using the Pythagorean Theorem. 7√2 7√2 14a) Make a prediction about how the sum sin 30 + cos 60 will relate to the sum sin 60 + cos 30. b) Use the sine and cosine values of special angles to find the sum: sin 30 + cos 60. c) Find the exact value of the sum: sin 60 + cos 30. d) Was your prediction correct ? Explain why or why not.