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1 c Marcia Drost, March 7, 2008 150 Lecture Notes - Section 6.3 Trigonometric Functions of Angles Definition of the Trigonometric Functions Let θ be p an angle in standard position and let P(x,y) be a point on the terminal side. If r = x2 + y 2 is the distance from the origin to the point P(x,y), then y r cos θ = x r tan θ = y (x 6= 0) x r csc θ = (y 6= 0) y sec θ = r (x 6= 0) x cot θ = x (y 6= 0) y sin θ = Reference Angle - Let θ be an angle in standard position. The reference angle θ̄ associated with θ is the acute angle formed by the terminal side of θ and the x-axis. Example 1 - Find the reference angle for each of the following: 5π (a) θ = 3 (b) θ = 870o Evaluating Trigonometric Functions Using Reference Angles To find the values of the trigonometric functions for any angle θ, follow these steps: 1. Find the reference angle θ̄ associated with the angle θ. 2. Determine the sign of the trigonometric function of θ. 3. The value of the trigonometric function of θ is the same, except possibly for sign, as the value of the trigonometric function of θ̄. Example 2 - Find the value of each of the following. (a) sin 240o (b) cot 16π 3 c Marcia Drost, March 7, 2008 2 Evaluating Trigonometric Functions Using Special Triangles An alternative method to finding the values of the trigonometric functions for any angle θ is as follows: 1. Find the reference angle θ̄ associated with the angle θ. (same as above) 2. Draw the appropriate special triangle using the reference angle: use the terminal side of θ as the hypotenuse and use the angle θ̄ as the acute angle of the right triangle with vertex at the origin. 3. Fill in the standard side lengths of the special triangle. Adjust the signs of the adjacent and opposite sides as appropriate to the quadrant the terminal side is in. The hypotenuse is ALWAYS positive. Example 3 - Find the value of each of the following. (a) cos 495o π (b) sec − 4 (c) tan 3π (d) sec 7π 2 3 c Marcia Drost, March 7, 2008 Pythagorean Identities sin2 θ + cos2 θ = 1 tan2 θ + 1 = sec2 θ 1 + cot2 θ = cscθ Let θ be an acute angle such that sin θ = 0.6 Find cos θ and tan θ. sin2 θ + cos2 θ = 1 (.6)2 + cos2 θ = 1 cos2 θ = .64 cos θ = .8 .6 .8 3 tan θ = = .75 4 tan θ = Let θ be an acute angle such that tan θ = 3 θ Find a) cot θ and b) sec θ Evaluate: sec (5o 40′ 12”) 40 12 5+ + = 5.67o 60 3600 sec (5.67o) = 1 ≈ 1.00492 cos 5.67o Reciprocal Identities 1 csc θ = sin θ tan θ = sin θ cos θ sec θ = 1 cos θ cot θ = cos θ sin θ cot θ = 1 tan θ Example 4 - Use the Pythagorean and reciprocal identities to express tan θ in terms of sin θ, where θ is in quadrant II. 4 c Marcia Drost, March 7, 2008 Example 5 - Given tan θ = 2 3 and θ in quadrant III, find cos θ. Area of a Triangle The area A of a triangle with sides of lengths a and b and with included angle θ is 1 A = ab sin θ 2 Example 6 - Find the area of a triangle with sides of length 7 and 9 and included angle 120o. Example 7 - Find the area of the shaded region in the figure below if the radius is 8 π inches, and the central angle is . 6