Download ExamView - Review 2.tst

Name: ________________________ Class: ___________________ Date: __________
ID: A
Review 2
Show All the steps to get full credit:
1. Verify the identity shown below.
cos θ (tan θ − sec θ )
= sin θ
1 − csc θ
2. Verify the identity shown below.
sec θ − sin θ tan θ = cos θ
3. Verify the identity shown below.
ÊÁ 2
ˆ
ÁÁ tan θ − 1 ˜˜˜ cot θ
ÁË
˜¯
= sec θ + csc θ
sin θ − cos θ
4. Verify the identity shown below.
cos α − cos β
sin α − sin β
+
=0
sin α + sin β
cos α + cos β
5. Verify the identity shown below.
ˆ˜
Áπ
2
2Ê
sec µ − cot ÁÁÁÁ − µ ˜˜˜ = 1
˜¯
Ë2
6. Verify the given identity.
2
2
cos(x + y) cos(x − y) = cos x − sin y
7. If sinx =
3
1
and cos x =
, evaluate the following function.
2
2
tanx
8. Use fundamental identities to simplify the expression below and then determine which of the
following is not equivalent.
sin α (csc α − sin α )
1
Name: ________________________
ID: A
9. Factor; then use fundamental identities to simplify the expression below and determine which of the
following is not equivalent.
cot 2 α tan2 α + cot 2 α
10. Multiply; then use fundamental identities to simplify the expression below and determine which of
the following is not equivalent.
(2 − 2 cos x ) (2 + 2 cos x )
11. If x = 10 cosθ , use trigonometric substitution to write
where 0 < θ < π .
12. If x = 6 sinθ , use trigonometric substitution to write
where 0 < θ <
π
2
100 − x as a trigonometric function of θ,
2
36 − x as a trigonometric function of θ,
2
.
13. Solve the given equation
sec x − 2 = 0
14. Solve the following equation.
2 sinx − 1 = 0
15. Solve the following equation.
sin2 x + sinx = 0
16. Solve the multiple-angle equation.
cos
3
x
=−
2
2
ÍÈ
ÍÎ 2
utility to approximate the angle x. Round answers to three decimal places.
Í π
17. Use the Quadratic Formula to solve the given equation on the interval ÍÍÍÍ 0,
18 tan2 x − 13 tanx + 2 = 0
2
ˆ˜
˜˜ ; then use a graphing
˜˜
¯
Name: ________________________
ID: A
18. Use the identities where needed to find all solutions (if they exist) of the given equation on the
interval [0, 2π ) .
cos 2 x − 5 sinx + 5 = 0
19. The horizontal distance d (in feet) traveled by a projectile with an initial speed of v feet per second is
modeled by
d=
2
v
sin2θ ,
32
where θ is the angle at which the projectile is launched.
Find the horizontal distance traveled by a golf ball that is hit with an initial speed of 50 feet per
second when the ball is hit at an angle of θ = 50° . Round to the nearest foot.
20. Find the exact value of the given expression.
ÁÊ 5π 5π ˜ˆ˜
˜
sinÁÁÁÁ
−
4 ˜˜¯
Ë 3
21. Find the exact value of the given expression.
cos (240° + 225° )
22. Find the exact value of the given expression using a sum or difference formula.
sin285°
23. Write the given expression as the sine of an angle.
sin105° cos 35° + sin35° cos 105°
24. Write the given expression as the tangent of an angle.
tan6x + tan2x
1 − tan6x tan2x
25. Find the exact value of sin(u + v ) given that sinu =
7
12
and cos v = − . (Both u and v are in
25
13
Quadrant II.)
26. Find the exact value of cos (u + v ) given that sinu =
Quadrant II.)
1
7
12
and cos v = − . (Both u and v are in
25
13
Name: ________________________
ID: A
27. Write the given expression as an algebraic expression.
sin(arcsinx + arccos x)
28. Simplify the given expression algebraically.
ÊÁ
π ˆ˜
sinÁÁÁ x − ˜˜˜˜
ÁË
2¯
29. Use the figure below to determine the exact value of the given function.
csc 2θ
30. Find the exact solutions of the given equation in the interval [0, 2π ) .
cos 2x + 3 cos x + 2 = 0
31. Use a double angle formula to rewrite the given expression.
6 cos 2 x − 3
32. Use a double-angle formula to find the exact value of cos 2u when sinu =
7
π
, where
<u<π.
25
2
33. Use the figure below to find the exact value of the given trigonometric expression.
sin
θ
2
15
36
(figure not necessarily to scale)
4
Name: ________________________
ID: A
34. Use the half-angle formulas to determine the exact value of the given trigonometric expression.
tan
3π
8
35. Use the half-angle formula to simplify the given expression.
1 + cos 4x
2
36. Use the product-to-sum formula to write the given product as a sum or difference.
12 sin
π
6
cos
π
6
37. Use the sum-to-product formulas to write the given expression as a product.
sin6θ − sin4θ
38. The range of a projectile fired at an angle θ with the horizontal and with an initial velocity of v 0 feet
1 2
v sin2θ where r is measured in feet. A golfer strikes a golf ball at 120 feet per
32 0
second. Ignoring the effects of air resistance, at what angle must the golfer hit the ball so that it
travels 140 feet? (Round answer to nearest angle.)
per second is r =
5
ID: A
Review 2
Answer Section
SHORT ANSWER
1. ANS:
PTS: 1
2. ANS:
OBJ: Verify identities
sec θ − sin θ tan θ =
1
sin θ
− sin θ ⋅
cos θ
cos θ
1
sin2 θ
=
−
cos θ cos θ
=
1 − sin θ
cos θ
=
cos θ
cos θ
2
2
= cos θ
PTS: 1
OBJ: Verify identities
1
ID: A
3. ANS:
PTS: 1
OBJ: Verify identities
2
ID: A
4. ANS:
PTS: 1
OBJ: Verify identities
3
ID: A
5. ANS:
PTS: 1
6. ANS:
OBJ: Verify identities
PTS: 1
7. ANS:
OBJ: Use sum and difference formulas to verify trig identities
tanx =
3
3
PTS: 1
8. ANS:
OBJ: Evaluate trig function given other trig values
1 − cot 2 α
PTS: 1
OBJ: Use fundamental identities to determine equivalent expression
4
ID: A
9. ANS:
sec 2 α
PTS: 1
10. ANS:
OBJ: Use fundamental identities to determine equivalent expression
4 − cos 2 x
PTS: 1
11. ANS:
OBJ: Use fundamental identities to determine equivalent expression
PTS: 1
12. ANS:
OBJ: Write algebraic expressions as trig functions with trig substitution
PTS: 1
13. ANS:
OBJ: Write algebraic expressions as trig functions with trig substitution
10 sinθ
6 cosθ
x=
5π
3
PTS: 1
14. ANS:
x=
π
6
+ 2nπ and x =
PTS: 1
15. ANS:
x = nπ and x =
PTS: 1
16. ANS:
x=
OBJ: Verify solutions to trig equations
5π
+ 2nπ , where n is an integer
6
OBJ: Solve trig equations
3π
+ nπ , where n is an integer
4
OBJ: Solve trig equations
5π
7π
+ 4nπ and
+ 4nπ , where n is an integer
3
3
PTS: 1
17. ANS:
OBJ: Solve multiple-angle equations
PTS: 1
18. ANS:
OBJ: Use the Quadratic Formula to solve trig equations
x = 0.219, 0.464
x=
π
2
PTS: 1
OBJ: Solve trig equations by factoring
5
ID: A
19. ANS:
77
PTS: 1
20. ANS:
3 +1
2 2
PTS: 1
21. ANS:
1−
OBJ: Find exact value of expression using sum formula
3
2 2
PTS: 1
22. ANS:
OBJ: Find exact value of expression using sum formula
− 3 −1
2 2
PTS: 1
23. ANS:
OBJ: Find exact value of expression using sum or difference formula
sin(140° )
PTS: 1
24. ANS:
OBJ: Rewrite an expression using a sum or difference formula
tan(8x )
PTS: 1
25. ANS:
OBJ: Rewrite an expression using a sum or difference formula
sin(u + v ) = −
204
325
PTS: 1
26. ANS:
cos (u + v ) =
PTS: 1
27. ANS:
OBJ: Find exact value of expression using sum or difference formula with constraints
253
325
OBJ: Find exact value of expression using sum or difference formula with constraints
1
PTS: 1
28. ANS:
OBJ: Write trig expressions as algebraic expressions
−cos x
PTS: 1
OBJ: Simplify trig expressions using addition and subtraction formulas
6
ID: A
29. ANS:
csc 2θ =
13
12
PTS: 1
30. ANS:
x=
OBJ: Find exact value of trig function from diagram
2π
4π
, π,
3
3
PTS: 1
31. ANS:
OBJ: Find exact solutions to trig equations involving multiple angles
3 cos 2x
PTS: 1
32. ANS:
cos 2u =
OBJ: Rewrite an expression as a double angle
527
625
PTS: 1
33. ANS:
OBJ: Find exact value of double angle given quadrant restraints
26
26
PTS: 1
34. ANS:
tan
3π
=
8
OBJ: Find exact value of expression using the half-angle formula
2 +1
PTS: 1
35. ANS:
OBJ: Find exact value of expression using the half-angle formula
cos 2x
PTS: 1
36. ANS:
6 sin
OBJ: Rewrite an expression with a product-to-sum formula
π
3
PTS: 1
37. ANS:
OBJ: Use the product-to-sum formula to rewrite products
PTS: 1
38. ANS:
OBJ: Use the sum-to-product formula to rewrite difference as a product
2 cos 5θ sin θ
9°
PTS: 1
OBJ: Solve problems dealing with multiple angle formula
7