Download Practice Problems for Midterm 3

MATH 1060-004, Fall 2012
PRACTICE PROBLEMS for Midterm 3
This is BY NO MEANS a reflection of what the actual test might look like. You should still work
through problems from the book. These are just a glimpse at the types of problems you might want to
practice and I might write .
The following formulas will be provided on the test : FORMULAS :
sin( A + B) = sin A cos B + cos A sin B
sin( A − B) = sin A cos B − cos A sin B
cos( A + B) = cos A cos B − sin A sin B
sin( A − B) = cos A cos B + sin A sin B
tan( A + B) =
tan A + tan B
1 − tan A tan B
tan( A − B) =
tan A − tan B
1 + tan A tan B
sin(2A) = 2 sin A cos A
tan(2A) =
2 tan A
1 − tan2 A
cos(2A) = cos2 A − sin2 A = 2 cos2 A − 1 = 1 − 2 sin2 A
2 tan A
1 − tan2 A
r
1 − cos( A)
A
sin( ) = ±
2
2
r
A
1 + cos( A)
cos( ) = ±
2
2
A
1 − cos A
sin A
tan( ) =
=
2
sin A
1 + cos A
tan(2A) =
1
(cos( A + B) + cos( A − B))
2
A±B
A∓B
sin A ± sin B = 2 sin(
) cos(
)
2
2
A+B
A−B
cos A − cos B = −2 sin(
) sin(
)
2
2
A+B
A−B
cos A + cos B = 2 cos(
) cos(
)
2
2
cos A cos B =
1
Instructor : Radhika Gupta
PART A - NO CALCULATOR
Solve the following problems WITHOUT using calculator.
1. Evaluate the following ( Use sum/difference/half-angle formula): cos( 19π
12 )
2. Given sin A = 15 , cos B =
√
− 24
5 ,
A, B are in second quadrant find the following :
(a) cos( A − B)
(b) cot( A + B)
3. Find the exact value of the following without using calculator :
tan 325◦ − tan 25◦
1 + tan 325◦ tan 25◦
PART B - Allowed to USE calculator
4. Verify the following :
tan(
π
1 − tan θ
− θ) =
4
1 + tan θ
5. Solve the following trigonometric equations for ALL x in [0, 2π ] :
(a) cos( x +
π
4 ) − cos( x
− π4 ) = 1
(b) sin(2x ) − cos( x ) = 0
(c)
cos(2x )
sin(3x )−sin x
=1
6. Verify the following (Use two different formulas for cos(2A) in the numerator and denominator):
tan2 A =
1 − cos(2A)
1 + cos(2A)
7. Review Exercises Chapter 6, Pg 483, Problem 32
8. Page 444, Problem 36
9. Find the unit vector in the direction of < 17, 19 >.
10. Given v =< −2, − 21 >, u =< 12 , −2 >, find the following vectors and sketch graphs for each :
(a) v − u
(b) 8v + 8u
11. Section 6.4 Problem 73
12. Find the direction angle of vectors :
(a) 3i + 5i
(b) 7i - 4i
13. True/False : Each section has a couple of True/False problems towards the end of the exercises,
practice those.
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Instructor : Radhika Gupta
NOTE : I made a correction in class about solving multiple angle trigonometric equations. Here it
is illustrated again.
PROBLEM : Find solutions for sin(4x ) = 0 for ALL x in [0, 2π ]
Then 4x = 0, π, 2π, 3π, 4π, 5π, 6π, 7π, 8π, 9π, . . .
3π 4π 5π 6π 7π 8π 9π
Thus x = 0, π4 , 2π
4 , 4 , 4 , 4 , 4 , 4 , 4 , 4 ,...
which is same as
5π 3π 7π
9π
x = 0, π4 , π2 , 3π
4 , π, 4 , 2 , 4 , 2π, 4 , . . .
Therefore all answers in the above list uptill 2π are answers to the problem asked.
If we look for possible values for 4x only in [0, 2π ] we actually miss some solutions in [0, 2π ].
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