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§6-6 RIGHT TRIANGLE TRIGONOMETRY In this section we consider the trigonometry of right triangles. The trigonometric functions of concern are the sine, cosine and tangent functions. Definition For a right triangle, the sine of an angle θ is the ratio of the length of the leg opposite θ opposite leg to the length of the hypotenuse: sin θ = . hypotenuse Definition For a right triangle, the cosine of an angle θ is the ratio of the length of the leg adjacent adjacent leg θ to the length of the hypotenuse: cos θ = . hypotenuse Definition For a right triangle, the tangent of an angle θ is the ratio of the length of the leg opposite leg opposite θ to the length of the leg adjacent θ : tan θ = . adjacent leg The mnemonic SOH CAH TOA is often used to recall the relationship of the sides of the right triangle to the trigonometric functions. Example 1 Find the sine, cosine and tangent of θ and α in the right triangle shown below. α 5 3 θ 4 Solution For θ the opposite leg has length 3 and the adjacent leg has length 4. Therefore... sin θ = opposite leg 3 = hypotenuse 5 cos θ = adjacent leg 4 = hypotenuse 5 tan θ = opposite leg 3 = adjacent leg 4 For α the opposite leg has length 4 and the adjacent leg has length 3. Therefore... sin α = opposite leg 4 = hypotenuse 5 cos α = adjacent leg 3 = hypotenuse 5 tan α = opposite leg 4 = adjacent leg 3 Copyright©2007 by Lawrence Perez and Patrick Quigley Example 2 If cos θ = x for the triangle below, then find the length of AC. A θ 3 B adjacent leg hypotenuse 3 x = AC ACx = 3 3 AC = x Solution cos θ = Identity 2 2 For any angle θ, sin θ + cos θ = 1 . Example 3 If sin θ = Solution sin θ + cos θ = 1 2 2 cos θ = 1 − sin θ 2 2 ⎛ 1⎞ cos θ = 1 − ⎝ 2⎠ 3 2 cos θ = 4 3 cos θ = ± 2 Identity For any angle θ, tan θ = Example 4 If cos θ = Solution C 2 1 then find the cosine of θ. 2 2 sin θ . cosθ 5 12 and tan θ = then find the sine of θ. 13 5 sin θ cos θ tan θ ⋅ cos θ = sin θ 12 5 ⋅ = sin θ 5 13 12 = sin θ 13 tan θ = Copyright©2007 by Lawrence Perez and Patrick Quigley §6-6 PROBLEM SET Find each quantity for the right triangle below. θ 13 5 α 12 1. sin α 2. cos α 3. tan α 4. sin θ 5. cos θ 6. tan θ Find the lengths of the missing sides of the right triangle below. y cos θ = x 2 3 tan θ = θ 5 2 10 7. Find x. 8. Find y. Use the given quantities to calculate the missing quantity. If sin θ = 1 3 and cos θ = then find tan θ. 2 2 10. If sin θ = 2 and tan θ = 1 then find cos θ. 2 11. If cos θ = 1 and tan θ = 2 2 then find sin θ . 3 12. If cos θ = 5 and sin θ > 0 then find sin θ . 3 9. Copyright©2007 by Lawrence Perez and Patrick Quigley §6-6 PROBLEM SOLUTIONS 1. 5 13 2. 12 13 3. 5 12 4. 12 13 5. 5 13 6. 12 5 7. 15 8. 5 5 9. 3 3 11. 2 2 3 12. 2 3 10. 2 2 Copyright©2007 by Lawrence Perez and Patrick Quigley