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Values of Trigonometric Functions (Sec. 6.4) Feature Details / Examples How are we really going to find values of trig functions? (And why bother with reference angles?) (1) “I know this angle is over 90°, but the stupid calculator keeps saying sin-1(.636) is 39.494°!” (2) “Mr. Byrd says he wants the exact value for tan(82π/6), and I have a feeling –1.732050808 isn’t exact!” Definition of reference angle Let θ be a nonquadrantal angle in standard position. The Review terms: nonquadrantal angle, terminal side Reference angles in each quadrant reference angle for θ is the acute angle θR that the terminal side of θ makes with the x-axis. With 0° < θ < 360° or 0 < θ < 2π: If θ is greater than 360° or less than 0°, first find the _____________ angle θ with 0° < θ < 360° or 0 < θ < 2π. Example: finding reference Find the reference angle θR for θ, and sketch both angles in standard position. angles (a) θ = 315° (b) θ = -240° (c) θ = 5π/6 (d) θ = 4 1 Theorem on Reference Angles If θ is a _______________ angle in standard position, then to find the value of a trigonometric function at θ, find its value for the _____________________ and prefix the appropriate sign. But what is the appropriate sign? Signs of Trigonometric Functions Functions that are positive in the quadrant shown: Mnemonic: “A Smart Trig Class” = All, Sin, Tan, Cos* II I Sin, csc ALL III IV Tan, cot Cos, sec . *Functions have the _______ sign as their reciprocal, so the mnemonic mentions only sin, tan, cos. Example: using reference angles Find the exact values of sin θ, cos θ, and tan θ if Finding the angle from the function value A common real-life situation: hypotenuse & height are known. Solution: inverse trigonometric functions. Example If sin θ = 0.6635, what is θ? (a) θ = 5π/6 (b) θ = 315° On calculator, find approx. value of sin-1 0.6635 Finding acute angle solutions of equations with a calculator “I know this angle is over 90°, but the stupid calculator keeps saying sin-1(.636) is 39.494°!” 2 Finding angles with a calculator Note: if cos θ is negative, θ is not the negative of the reference angle! DAB, March 2011 3