Download Math 104 Sample Challenge Exam

Mesa College – Math 104 ( Trig) Challenge Exam SAMPLES
Directions: NO CALCULATOR. Write neatly and show your work and steps. Answers without appropriate work shown will receive
little or NO credit. Be sure to simplify all radicals and fractions. Attach your neat and organized solution sheets behind this cover
sheet. Make sure each solution is properly labeled.
#1 – 3. Use the given information to find the values of the remaining Trigonometric functions.
1. If sin θ = −
cos θ =
1
and
2
2. If sin θ =
3. If cot θ = −4 and
3
and θ is in QII,
5
270o ≤ θ < 360o
3
2
#4-6 . Find ALL values of x. Express your answers using radians.
4.
3csc x − 6 = 0.
5.
tan 2 x = − tan x
6.
2 sin 2 x − 5sin x − 3 = 0
7. Find the side length x in each of the triangles shown.
(7b)
(7a)
10
60 °
40 °
x
8. If cos x = 0.8 and sin x = −0.6 , find :
(8a) sin(2 x )
9. Given that cos α =
(8b) cos(2 x )
(8c) in which quadrant is angle (2x) ?
2
1
−π
π
and sin β = − , where
≤ β ≤ and 0 ≤ α ≤ π find :
3
4
2
2
(9a) sin (α + β )
α 

2
(9b) cos(α − β )
(9c) cos 
(9d) in which quadrant is angle ( α + β )?
(9e) in which quadrant is angle ( α − β )?
α 
?
2
(9f) in which quadrant is angle 
10. Find the principal value of each:
− 3
 2 


10a) Arc cos 


10b) sin −1  −
1

2
10c)
 3
tan −1 

 3 
10d) arc cot
( 3)


10e) cos  arcsin  3



2  


10f) cos ( arctan(−1) )
10g) cot −1  sin
3π 

2 
11. Find the exact values of each:
11a)
π 
sin  
4
 −3π 

 4 
( )
 13π 

 6 
11e) cot 
(
11d) sec −420o
)
11b) cos 30o
11c) csc 
π 
sin  
6
11f)
π 
1 + cos  
6
11g) cos 45o cos 60o − sin 45o sin 60o
( ) ( )
( ) ( )
12. Simplify each expression completely.
o
 3π 
2  3π 
 − cos  
 8 
 8 
 5π 

 8 
(12b) sin 2 
o
(12a) sin(75 ) cos(15 ) − cos(75) sin(15)
(12c) cos 
13. Find all solutions between 0 ≤ θ < 360o for each:
 x
2
(13a) cos 2 θ − 2sin θ = −2
(13c) sin(2 x) = cos( x)
(13b) csc   = 2
14. Sketch at least one period of the graph of each. Label the coordinates of one maximum point and the coordinates of
one minimum point.


(14a) f ( x) = −2 cos  x +
15. Simplify the expression
(a) cos y
π


 +1
2
(14b) g ( x) = 5sin 2  x −
π
(14c) h( x) = 3 + 2sec x

4
cot y − 1
, so that it matches one of the expressions below. SHOW STEPS.
1 − tan y
(b) tan y
(c ) 0
(d)
sec y
csc y
(e) cot y
16. Simplify each expression.
(16a) (sec β − tan β )(sec β + tan β )
(16b)
cos θ
1 + sin θ
+
1 + sin θ
cos θ
17. Complete the trigonometric identity: sec θ − cos θ = w , by selecting w from the list of expressions below.
(a) tan θ sin θ
(b) cot 2 θ cos θ
(c )
1 − cos θ sin θ
sin θ
(d)
sec θ − 1
sec θ
18. SIMPLIFY each to a single trig expression involving no fractions.
(18a) 1 +
tan 2 x
1 + tan 2 x
(18b)
( sin ( ) + cos ( ) )
x
2
x
2
2
−1
(e) -1
19. If A = 3 + 2i and B = 5 − i , where i = −1 , find each:
(19a) A2
(19b) AB
20. Evaluate: 2i 2 − 5i 75 +
(19c)
A
B
6
, where i = −1 .
i13
21. Express each in standard form (i.e. a ± bi )


(21a) 5 2  cos
( ) − i sin 
5π
5π
4
4



(21b) 8cis30o
22. Convert each from Polar coordinates to Cartesian coordinates
(22a)
( 2, 30 )
o


(22b)  −6,
4π 

3 
23. Convert each from Cartesian coordinates to Polar coordinates
(
(23a) 3, 3
)
(23b) ( −5, −5)
4
24. If 4 + 3i ≈ 5cis37 o , find ( 4 + 3i ) in trigonometric form.
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Answers to Math 104 (Trig) SAMPLES
{ This is a DRAFT… If you find any errors, please contact me at: [email protected] … Thanks. }
1. sin θ =
−1
3
2 3
− 3
, cos θ =
, csc θ = −2 , sec θ =
, tan θ =
, cot θ = − 3
2
2
3
3
2. sin θ =
3
−4
5
−5
−3
−4
, cos θ =
, csc θ = , sec θ =
, tan θ =
, cot θ =
5
5
3
4
4
3
3. sin θ =
− 17
4 17
17
−1
, cos θ =
, csc θ = − 17 , sec θ =
, tan θ =
, cot θ = −4
17
17
4
4


π


7π
11π

+ 2kπ or x =
+ 2k π 
6
6

4.  x | x =
6.  x | x =
6
+ 2kπ or x =
5π

+ 2 kπ 
6



5.  x | x = kπ or x =
7a. x = 10 39
3π

+ kπ 
4

7b. x =
20sin 80o
3
8a.
−24
= − 0.96
25
8b.
9c.
+ 30
6
9d. 9e. 9f. all are in Q-I
10e.
1
2
10f.
11c. − 2
13b.
2
2
2
2
12b.
{90 , 270 }
o
o
15. E
16a. 1
18a. cos 2 x


π


5π 

4 
23b.  5 2,
12c.
2− 2
2
{

6
24. 625cis (148o )
5 3−2
12
5π
6
10b.
9b.
−π
6
2
2
11a.
10c.
{ }
14. GRAPHS… see below
16b. 2sec θ
17. A
19a. 5 + 12i
1 1
+ i
2 2
21b. 4 3 + 4i
20. −2 − i
22a.
(
)
3,1
(
22b. 3, 3 3
π
6
2− 6
4
11g.
13a. 90o
}
2 15 − 5
12
3
2
11b.
11f. 2 − 3
3
13c. 90o , 270o , 30o ,150o
19c.
21a. −5 + 5i
9a.
10a.
11e.
18b. sin x
19b. 17 + 7i
23a.  2 3,
8c. Q-IV
10g. undefined.
11d. 2
3
2
12a.
7
= 0.28
25
)
10d.
π
6