Applications of Trigonometry, calculus homework help

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1.) Find the length a. 2.) Find the missing side length. For questions 3 – 4 assume the triangle has the given measurements. Solve for the remaining sides and angles. 3.) C  100 , a  4.) A  1 1 , b 2 3  , b  17.4, c  19.6 6 For questions 5 – 6 solve for the remaining angles. 5.) 6.) 7.) Viewed from Earth, two stars form and angle of 83.23°. Star A is 33 light-years from Earth, and Star B is 21 light-years from Earth. Sketch a diagram modeling this situation and find out how many light-years the stars are away from each other. Questions 8 – 10 deal with the ambiguous SSA case. For each, find all possible solutions and sketch the triangle in each case. 8.) A  50 , a  5, b  3 9.) A  50 , a  3, b  5 10.) A  130 , a  1.2, b  3 11 – 14. Solve the given triangles by finding the missing angle and other side lengths. 11.) 12.) 13.) 14.) Compute the following linear combinations. 15.) 4 1, 7  2 5,  3  16.)  22.2, 19.9  3  0.8, 6.3 17.) Let v1   3, 5 and v2   4, 7 . (a) Compute v1 and v2 (b) Compute the unit vectors in the direction of v1 and v2 . (c) Draw and label v1, v2 , and their unit vectors on the axes provided. 18.) Let v1   6, 4 and v2   3, 6  . Compute the following. (a) v1 • v2 (b) The angle between v1 and v2 . (c) The scalar projection of v1 onto v2 . (d) The projection of v1 onto v2 . 19.) 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 20.) Use the trigonometric subtraction formula for sine to verify this identity:   sin   x   cos x 2   Answer: Calculation Reason 21.) Find the exact value of each of the following using half-angle formulas: (a)  7  cos    12   17  tan    12    cot 15  sin 195 Answer: 22.) Rewrite sin4 x so that it involves only the first power of cosine. Answer: 23.) Prove the identity: sin  2 x  sin x Answer:  cos  2 x  cos x  sec x 24.) Prove the identity: cos x  cos y 2   sin x  sin y 2  2  2cos  x  y  Answer: 25.) Solve these equations graphically on the interval 0,2 . Sketch the graph and list the solutions. (a) sin x  1  cos x sin  2x   1  tan x Answer: 26.) Solve these equations: (a)     sin  x    sin  x    1 4 4     (b) sin x cos x  (c) Answer: 3 4 tan2 x  3 tan x  2  0
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