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Name______________________ Date__________________ Algebra II/Trig Regents Review #11: Trig Equations and Graphs Trig Identities: Reciprocal Identities: Quotient Identities: sec 1 cos tan sin cos csc 1 sin cot cos sin cot 1 tan Examples: 1. (cos )(tan ) 2. (sec )(cot ) 3. csc x sec x Example Regents Questions 1. Prove that the equation shown below is an identity for all values for which the functions are defined: csc sin 2 cot cos Pythagorean Identities: Given the Pythagorean identities cos 2 sin 2 1 , you can derive four other identities: Examples: Write each expression in terms as a single expression: 1. 2. tan cot 1 sin 2 Example Regents Questions 1. The expression 2 (1) cos 2. The expression 2 (1) 1 cos sin 2 cos 2 is equivalent to 1 sin 2 2 (3) sec 2 (2) sin cos2 sin is equivalent to sin 2 (2) cos 2 (4) csc (3) sin (4) csc 2 2 2 3. Starting with sin A cos2 A 1, derive the formula tan A 1 sec A. Sum/Difference of two angles: sin( A B) sin A cos B cos A sin B cos( A B) cos A cos B sin A sin B tan( A B) Example: Find the exact value of tan A tan B 1 tan A tan B sin105 Find the exact value of cos15 Example Regents Questions 1. The expression cos4xcos3x sin 4xsin 3x is equivalent to (1) sin x 2. (3) cosx cos2 cos2 is equivalent The expression to 2 (1) sin (2) sin 7x 3. If tan A 2 (2) sin 2 (3) cos 1 (4) cos 7x 2 (4) cos 1 2 5 and sinB = and angles A and B are in Quadrant I, find the value of tan( A B). 3 41 Double angle formulas: Use the following formulas for double angles sin 2 A 2sin A cos A cos 2 A cos 2 A sin 2 A cos 2 A 2 cos 2 A 1 cos 2 A 1 2sin 2 A tan 2 A 2 tan A 1 tan 2 A Examples: If cos 7 , find cos 2 . 25 If sin 4 , find sin 2 . 5 Express as a single trig function: sin 2 2 cos Example Regents Questions 1. If sin A (1) 2. 2 5 3 If cos y 2 where 0 o A 90 o, what is the value of sin 2A? 3 (2) 2 5 9 (3) 4 5 9 4 and angle y is an acute angle, what is the value of cos2y ? 5 (4) 4 5 9 Finding the measure of an angle: When finding all values of theta when given the value of the trig function, follow these steps: 1. Find the reference angle by pressing, 2nd -- trig function—value Always plug in a positive value The answer you find is a reference angle, not an answer!! 2. Use the sign of the given value to determine the quadrants in which your angle lies 3. Find the measure of the angles Examples: 1. sin 4 5 2. cos 9 10 Solving trig equations: The 4 steps for solving trig equations are: 1. 2. 3. 4. 5. Isolate the trig equation Find the reference angle Determine the quadrants Find the angle Check Examples: 1. 2cos 3 4 3. 4csc 5 3csc 4 2. 3tan 2 1 3. csc 1 2 Solving Second Degree Trig Equations: You will be required to factor just as you would factor with x as the variable!! Examples: 1. sin 2 sin 1 1 3. 2cos 2sin cos 6. cos 2x cos x 2. 2 cos 2 cos 4. cos2 3sin 0 Graphing Trigonometric Functions: y sin x y cos x y tan x y a sin(bx c) d y a cos(bx c) d Amplitude: a _________________________________________________________________________________ Frequency: b__________________________________________________________________________________ Period: 2 ___________________________________________________________________________________ b Phase Shift: c ________________________________________________________________________________ Reflection: ____________________________________________________________________________________ Vertical Shift: d________________________________________________________________________________ Examples Regents Questions 1. What is the period of the function y (1) 1 2 (2) 1 x sin 2 3 1 3 (3) 2 3 (4) 6 2. Which graph represents one complete cycle of the equation y sin 3x? 3. A radio transmitter sends a radio wave from the top of a 50-foot tower. The wave is represented by the accompanying graph. What is the equation of the radio wave? (1) y sin x (2) y 1.5sin x (3) y sin1.5x (4) y 2sin x 4. What is the period of the function f ( ) 2cos3 ? (1) 2 (2) 3 3 (3) 2 (4) 2 5. Which equation is graphed in the diagram below? x 8 30 x 8 30 (1) y 3cos x 5 15 (3) y 3cos (2) y 3cos x 5 15 (4) y 3cos 6. The accompanying graph shows a trigonometric function. State the equation of this function. 7. A radio wave has an amplitude of 3 and a wavelength (period) of (a) Write an equation for this function. meters. (b) On the accompanying grid, using the interval 0 to 2 , draw a possible sine curve for this wave that passes through the origin. 1 2 8. (a) On the same set of axes, sketch the graphs of y 2cos x and y = -sinx in the interval 0 x 2 . (b) Give the exact value(s) of the coordinates of the intersection points of the two graphs.