Complementary Angles Investigation Name ________________________________________________________________ Period _________ Complementary Angles Investigation Materials: Straightedge, graph paper, patty paper, and protractor. No calculator! Part 1: On the grids below, and using your ruler, draw two different right triangles. Label each of the angles A, B, and C so that angle A and B are acute angles, and angle C is 90˚. Measure the sides and label them as well. Triangle #1: Triangle #2: What relationship do angles A and B have? What name do we give special angles like this? Find the sine, cosine, and tangent for both triangles for both angles A and B. What relationship do you notice between trigonometric ratios of the complementary angles? Triangle 1: Sin A = ______ Sin B = _____ Cos A = _____ Cos B = _____ Tan A = _____ Tan B = _____ Triangle 2: Sin A = ______ Sin B = _____ Cos A = _____ Cos B = _____ Tan A = _____ Tan B = _____

Part 2: Get into groups of 3-4 and compare your results (remember that your triangles may be different than other group member’s). Organize your data below. Part 3: Use your understanding of the relationship between trig ratios of complementary angles to complete the following problems. (Note that these numbers are not accurate, so using your calculator will NOT help you!) Given the triangle and trig ratios below, find sin(β) and cos(β). Suppose that sin(19˚) = 0.2 and cos(19˚) = 0.98. Find sin(71˚) and cos(71˚). Suppose that sin(34˚) = 0.35 and cos(θ) = 0.35. Find θ. α β sin(α) = ⅓ cos(α) = ¼

Part 4: Prove that the relationship between trigonometric ratios of the complementary angles holds true for the triangle below. B A C a b c Write a statement that generalizes the relationship between between the sine and cosine ratios of complementary angles. Did you notice any other interesting things about trig ratios that you’d like share?

y = sin(x) y = cos(x) Part 5: Working individually, carefully examine the two graphs below. Explain in two different ways how these graphs demonstrate the same relationship we’ve discovered between sine and cosine functions and complementary angles. Get back into your groups and prepare to share your ideas. y = sin(x) 90˚ 180˚ 270˚ -270˚ -180˚ -90˚ 1 -1 y = cos(x) 90˚ 180˚ 270˚ -270˚ -180˚ -90˚ 1 -1