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applications_of_trigonometry.docx

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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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