Trigonometric Identities, calculus homework help

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For questions 1 – 5, decide whether the equation is a trigonometric identity. Explain your reasoning.   cos2 x 1  tan2 x  1   sec x tan x 1  sin2 x  sin x   csc x 2sin x  2  0 cos2  2x   sin2  2x   0 sin2  csc 2   sin2   cos2  For questions 6 – 9 prove the following trigonometric identities. sec   csc  tan  Calculation Reason 1  sin x   1  cos x   sec x  tan x   Calculation Reason    sec   x  6 3 cos x  sin x   2 Calculation Reason sin   Prove tan    for  in quadrant 1  2  1  cos  Calculation Reason For questions 10 – 14 solve the following equations: sin2 x  2cos x  2 2sin  2x   2sin x  2 3 cos x  3  0 csc  4x   2  0 sin  2x   sin x Consider the equation sin2 x cos2 x  (a) Show that sin2 x cos2 x  Calculation 1  cos 4 x 8 1  cos 4 x by filling in the table below. Start with the left hand side. 8 Reason Calculation Reason (b) Find all solutions to the equation sin2 x cos2 x  2 2 . 16 Use the half-angle formulas to come up with an exact expression for each function value below. You do not have to simplify your answers. (a)  tan   8   (b) cos   8   tan    16  (c) Recall that the equation for horizontal distance h in feet of a projectile with initial velocity v 0 and initial v 2  angle  is given by h   0  sin  cos  .  16    (d) Assume the initial velocity is 60 feet / second. What initial angle will you need to ensure that the horizontal distance will be exactly 100 feet? (e) Assume the initial velocity is 60 feet / second. What is the maximum horizontal distance possible, and at what angle does this occur? 17.) Derive this identity from the sum and difference formulas for cosine: sin a sin b  1 cos  a  b   cos  a  b  2 Start with the right-hand side since it is more complex. Answer: Calculation Reason 1. Use the trigonometric subtraction formula for sine to verify this identity:   sin   x   cos x 2  Answer: Calculation Reason 2. Find the exact value of each of the following using half-angle formulas: a.) b.) c.) d.) Answer:  7  cos    12   17  tan    12    cot 15  sin 195 4 3. Rewrite sin x so that it involves only the first power of cosine. Answer: 4. Prove the identity: sin  2 x  sin x  cos  2 x  cos x  sec x Answer: 5. Prove the identity: cos x  cos y 2   sin x  sin y 2  2  2cos  x  y  Answer: 22.) Solve these equations graphically on the interval 0,2 . Sketch the graph and list the solutions. (c) sin x  1  cos x (d) sin  2x   1  tan x Answer: 23.) Solve these equations: (e)     sin  x    sin  x    1 4 4   (f) sin x cos x  3 4 (g) tan2 x  3 tan x  2  0 Answer:
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