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CC Geometry H Aim #27: Students rewrite the Pythagorean theorem in terms of sine and cosine ratios and write tangent as an identity in terms of sine and cosine. Do Now: 1) In a right triangle, with acute angle θ, sin θ = 1/2. What is the value of cos θ? θ 2) In a right triangle, with acute angle θ, sin θ = 7/9. What is the value of tan θ? θ 3) What common Pythagorean triple was probably modeled in the construction of the triangle in figure 2? Use sin 53o ≈ 0.8. The Pythagorean Identity -Let θ be the angle such that sin θ = 0.8. Above, we used an approximation. We do not know exactly what the angle measure is, but we will assign the angle measure that results in the sine of the angle as 0.8 the label θ. We have the following triangle: 0.8 1 θ 0.6 -Since the value of sin θ is 0.8 and the value of cos θ is 0.6, how can we rewrite the leg lengths? 1 θ Apply the Pythagorean Theorem to this triangle: 1 sin θ θ cos θ The statement above is called the Pythagorean Identity. To prove this identity, we will use this diagram: opp2 + adj2 = hyp2 hyp opp θ Divide both sides by hyp2: adj The trigonometric identity sin θ Prove that tan θ = cos θ sin θ = c a cos θ = θ b tan θ = If you are given one of the values of sin θ, cos θ or tan θ, we can find the other θ two values using the identities (sin θ)2 + (cos θ)2 = 1 and tan θ = sin or by using cos θ the Pythagorean theorem. Applying the Pythagorean Identity: 1) In a right triangle, with acute angle θ, sin θ = 1/2. Use the Pythagorean identity to determine the value of cos θ. 2) In a right triangle, with acute angle θ, sin θ = 7/9. Use the Pythagorean identity to determine the value of tan θ. 3) If cos β = 2/3, use the Pythagorean identity and the trigonometric identity to find sin β and tan β. 4) Find the missing side lengths of the following triangle using sine, cosine, and/or tangent. Round your answer to the nearest ten-thousandth. 5) The right triangle shown is taken from a slice of a right rectangular pyramid with a square base. a. Find the height of the pyramid to the nearest tenth. b. Find the lengths of the sides of the base of the pyramid to the nearest tenth. c. Find the lateral surface area of the right rectangular pyramid. (lateral surface area: the area of the side faces, not including the base. 6) A machinist is fabricating a wedge in the shape of a right triangular prism. One acute angle of the right triangular base is 33o, and the opposite side is 6.5 cm. Find the length of the edges labeled l and m using sine, cosine, and/or tangent. Round your answer to the nearest thousandth of a centimeter. adj Let's Sum It Up • The Pythagorean Identity is (sin θ)2 + (cos θ)2 = 1. opp • Tangent can be represented as tan θ = sin θ or tan θ = adj cos θ . • If you have one of the values of sin θ, cos θ, or tan θ, you can use the above identities or the Pythagorean Theorem. Name_____________________ Date _____________________ 1) If cos β = 2) If sin θ = CC Geometry H HW #27 , use trigonometric identities to find sin β and tan β. , use trigonometric identities to find cos θ and tan θ. 3) If tan θ = 5, use the Pythagorean theorem to find the hypotenuse, and then find cos θ and sin θ. θ 4) If sin θ = , use trigonometric identities to find cos θ and tan θ. 5) Find the missing side lengths of the following triangle using sine, cosine, and/or tangent. Round your answer to the nearest ten-thousandth. OVER 6) A surveying crew has two points A and B marked along a roadside at a distance of 400 yd. A third point C is marked at the back corner of a property along a perpendicular to the road at B. A straight path joining C to A forms a 28 degree angle with the road. Find the distance from the road to point C at the back of the property and the distance from A to C using sine, cosine, and/or tangent. Round your answer to the nearest thousandth. Review: 1) In quadrilateral ABCD, the diagonals bisect its angles. If the diagonals are not congruent, quadrilateral ABCD must be a _______________. 2) In the diagram below, QM is a median of triangle PQR and point C is the centroid of triangle PQR. If QC = 5x and CM = x + 12, determine and state the length of QM. 3) Use a compass and straightedge to divide line segment AB below into four congruent parts. [Leave all construction marks.]