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Trigonometric Functions of Any Angle MATH 109 - Precalculus S. Rook Overview • Section 4.4 in the textbook: – Trigonometric functions of any angle – Reference angles – Trigonometric functions of real numbers 2 Trigonometric Functions of any Angle Trigonometric Functions of Any Angle • Given an angle θ in standard position and a point (x, y) on the terminal side of θ, then the six trigonometric functions of ANY ANGLE θ are can be defined in terms of x, y, and the length of the line connecting the origin and (x, y) denoted as r 4 Trigonometric Functions of any Angle (Continued) Function Abbreviation Definition y r x The cosine of θ cos θ r y The tangent of θ tan θ ,x 0 x x The cotangent of θ cot θ ,y0 y r The secant of θ sec θ ,x 0 x r The cosecant of θ csc θ ,y0 y Where r x 2 y 2 and x and y retain their signs from (x, y) The sine of θ sin θ 5 Trigonometric Functions of any Angle (Continued) Function Abbreviation Definition y r x The cosine of θ cos θ r y The tangent of θ tan θ ,x 0 x x The cotangent of θ cot θ ,y0 y r The secant of θ sec θ ,x 0 x r The cosecant of θ csc θ ,y0 y Where r x 2 y 2 and x and y retain their signs from (x, y) The sine of θ sin θ 6 Algebraic Signs of Trigonometric Functions • The sign of the six trigonometric functions depends on which quadrant θ terminates in: r is the distance from the origin to (x, y) so it is ALWAYS positive – The signs of x and y depend on which quadrant (x, y) lies – Remember the shorthand notation involving “the element of” symbol: • i.e. QIV means theta is a standard angle which terminates in Q IV 7 Algebraic Signs of Trigonometric Functions (Continued) Functions sin y r cos x r y tan x θ Є QI θ Є QII θ Є QIII θ Є QIV and csc r y + + – – and sec r x + – – + and cot x y + – + – 8 Trigonometric Functions of any Angle (Example) Ex 1: Find the value of all six trigonometric functions if: a) (-1, 2) lies on the terminal side of θ b) (-7, -1) lies on the terminal side of θ 9 Trigonometric Functions of any Angle (Example) Ex 2: Given sec θ = -3⁄2 where cos θ < 0, find the exact value of tan θ and csc θ 10 Reference Angles Reference Angles • An important definition is the reference angle – Allows us to calculate ANY angle θ using an equivalent positive acute angle • We can now work in all four quadrants of the Cartesian Plane instead of just Quadrant I! • Reference angle: denoted θ’, the positive acute angle that lies between the terminal side of θ and the x-axis θ MUST be in standard position 12 Reference Angles Examples – Quadrant I Note that both θ and θ’ are 60° 13 Reference Angles Examples – Quadrant II 14 Reference Angles Examples – Quadrant III 15 Reference Angles Examples – Quadrant IV 16 Reference Angles Summary • Depending in which quadrant θ terminates, we can formulate a general rule for finding reference angles: – For any positive angle θ, 0° ≤ θ ≤ 360°: • If θ Є QI: θ’ = θ • If θ Є QII: θ‘ = 180° – θ • If θ Є QIII: θ‘ = θ – 180° • If θ Є QIV: θ’ = 360° – θ 17 Reference Angles Summary (Continued) – If θ > 360°: • Keep subtracting 360° from θ until 0° ≤ θ ≤ 360° • Go back to the first step on the previous slide – If θ < 0°: • Keep adding 360° to θ until 0° ≤ θ ≤ 360° • Go back to the first step on the previous slide – If θ is in radians: • Either replace 180° with π and 360° with 2π OR • Convert θ to degrees 18 Reference Angles (Example) Ex 3: i) draw θ in standard position ii) draw θ’, the reference angle of θ: a) 312° c) 4π⁄5 e) 11π⁄3 b) π⁄8 d) -127° 19 Trigonometric Functions of Real Numbers Reference Angle Theorem • Reference Angle Theorem: the value of a trigonometric function of an angle θ is EQUIVALENT to the VALUE of the trigonometric function of its reference angle – The ONLY thing that may be different is the sign • Determine the sign based on the trigonometric function and which quadrant θ terminates in – The Reference Angle Theorem is the reason why we need to memorize the exact values of 30°, 45°, and 60° only in Quadrant I! 21 Evaluating a Trigonometric Function Exactly • To evaluate a trigonometric function of θ: – Ensure that 0 < θ < 2π when using radians or 0° < θ < 360° when using degrees – Find θ’ the reference angle of θ – Evaluate the function using the EXACT values of the reference angle and the quadrant in which θ terminates • Write the function in terms of sine or cosine if necessary 22 Evaluating a Trigonometric Function (Exactly) Ex 4: Give the exact value: a) sin 225° b) cos 750° c) tan 120° d) sec -11π⁄4 23 Summary • After studying these slides, you should be able to: – Calculate the trigonometric function of ANY angle θ – State the reference angle of an angle θ in standard position – Evaluate a trigonometric function using reference angles and exact values • Additional Practice – See the list of suggested problems for 4.4 • Next lesson – Graphs of Sine & Cosine Functions (Section 4.5) 24