Download PART I A. Find the exact values of the following without a calculator

PART I
A. Find the exact values of the following without a calculator:
1. sin 15°
π
2. cos
8
3. 1− 2sin 2 15°
4. tan(75°)
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B. Simplify each of the following:
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1. 2sin(x + 20°)cos(x + 20°)
2. 1− 2sin 2 14 π
( )
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3. (sin x − cos x )
2
C. Find the values of sin 2x, cos 2x, and tan 2x in each of the following without a
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calculator.
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3
5
12
2. cos x =
13
3
3. tan x = −
4
1. sin x =
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€ If 3sin2x = 2cos x , find the values of cos 2x.
D.
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7
1
1
1
E. If cos x = − , find sin x, cos x, and tan x without a calculator.
25
2
2
2
F. Simplify
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cos2x + 1
cos x
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PART II
A. Solve the following trigonometric equations (no calculators). If no domain is given,
give the general solution (all solutions). If the domain is given in degrees, give the
answers in degrees. If the domain is given in radians, give the answers in radians.
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1.
2.
3.
4.
5.
6.
7.
8.
2cos x = 1
tan x + 1 = 0, 0 ≤ x ≤ 360°
4 sin x + 3 = 0, − π ≤ x ≤ π
2cos2 x + 3cos x − 2 = 0, 0 ≤ x ≤ 2π
2cos2 x − sin x = 1, – 180° ≤ x ≤ 180
3 cos x − sin x = 0, − π ≤ x ≤ π
cos2x − cos x = 0, 0 ≤ x ≤ 360°
2sin(x − 60°) = 1, 0 ≤ x ≤ 360°
⎛
1 ⎞
9. 3cos⎜ x − π ⎟ + 1 = 0, − π ≤ x ≤ π
⎝
6 ⎠
10. 3sin 3x + 1 = 0, 0 ≤ x ≤ π
11. tan 4 x = 3, 0 ≤ x ≤ 180°
1
12.
+ tan x = 4, − π ≤ x ≤ π
tan x
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B.€Simplify the following:
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sin x cos x
+
csc x sec x
1
1
2. cos2 1 + sin 2 1
2
2
2
3. (sin A + cos A) + (sin A − cos A) 2
1.
C. Prove the following whenever both sides have meaning:
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1.
2.
3.
4.
5.
6.
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7.
€
8.
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9.
2cot 2 x − 3csc 2 x = −2 − csc 2 x = −3 − cot 2 x
sin Asec Acot A + cos Acsc Atan A = 2
cot 2 Asec 2 A + tan 2 Acsc 2 A = sec 2 A + csc 2 A
2tan 2 A − 3sec 2 A = 1− 4 sec 2 A + 3tan 2 A
(cos A − sin A + 1) 2 = 2(1+ cos A)(1− sin A)
sin x
1+ cos x
+
= 2csc x
1+ cos x
sin x
2cos A
tan A + 1
+1=
sin A − cos A
tan A −1
1+ sec A
2
= −(cot A + csc A)
1− sec A
(1− sin A)(1− cot A)
= cos A
(1− csc A)(1− tan A)
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D. Expand, and/or simplify where possible. One step is enough in cases where an
expansion would be extremely long (such as #1).
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1. cos 3(A + B)
⎛π
⎞
2. sin⎜ + 2A⎟
⎝3
⎠
1
tan π + tan A
4
3.
1
1− tan π tan A
4
tan80° − tan 35°
4.
tan80° + cot 35°
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E. Express as a single trigonometric function:
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1. cos55°cos 35° − sin55°sin 35°
1
1
2. sin A −
3 cos A
2
2
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4
F. If A is an angle in quadrant 2 and B is an angle in quadrant 3 such that sin A = and
5
€
8
tan B = , evaluate cos(A – B), sin(A + B), and cos(A + B) without a calculator.
15
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G. Prove, whenever both expressions have meaning:
1. sin(a + b) + sin(a − b) = 2sin acosb
2. sin B + cos Asin(A − B) = sin Acos(A − B)
⎛π
⎞ cos A − sin A
⎛π
⎞
= cot⎜ + A⎟
3. tan⎜ − A⎟ =
⎝4
⎠ cos A + sin A
⎝4
⎠
tan 3A + tan A
4.
= 2cos2A
tan 3A − tan A
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H.
€
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1
1
1
1. If cos A = − , find sin A and cos A .
3
2
2
1
2. Find sin x, cos x, and tan x, if cos2x =
8
1
3. Prove that€tan π = 2€−1
8
€ without graphing:
I. Solve the following equations
€ 1. cos x − 2sin x = 1
2. 3sin x − 5cos x = 4
3. 6sin x + 3cos x = 5
€
For #4–5€solve in two ways: (i) by expressing cos 2x and sin 2x in terms of
t = tan x, and (ii) using an expression of the form Asin(2x ± α ) or Acos(2x ± α ) .
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4. 2cos2x + sin2x = 2
1
5. 2sin2x − 3cos2x = €
2
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PART III
€ where possible. Give exact answers. If a problem requires a calculator in
A. Evaluate
order to solve, then state that a calculator is required, and then evaluate it. Only one
problem requires a calculator.
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⎛ 1 ⎞
3⎟
1. arcsin ⎜−
⎝ 2 ⎠
⎛
3⎞
2. arccos⎜ arcsin ⎟
⎝
5⎠
3. arctan(tan(−π))
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⎛
⎛ 1 ⎞⎞
4. cos⎜arcsin⎜ − ⎟⎟
⎝ 2 ⎠⎠
⎝
B.
1. For which numbers a is it true that arcsin(sin a) = a?
2. For which numbers b is it true that sin(arcsin b) = b?
3. For which numbers c is it true that arcos(cos c) = c?
4. For which numbers d is it true that cos(arcos d) = d?
In all cases, explain your reasoning.
⎛1
⎞
1
C.
Using the identity cos⎜ π − θ ⎟ = sin θ , show that arcsin x + arccos x = π for all
⎝2
⎠
2
x ∈ [−1, 1] .
D.
1
€1
Find the €
exact value of arctan + arctan
2
3
E.
1
1 1
Prove that 2arctan + arctan = π
3
7 4
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PART IV
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A. Differentiate each of the following with respect to x.
1. arcsin(3x)
2. arcos(x + 1)
3. arctan x 2
( )
4. ln(arcsin x )
( )
5. x arctan x 2
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€ For the function f (x) = arctan2x ,
B.
1.
Sketch the graph of y = f (x)
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⎛ 1⎞
2.
Evaluate f ⎜ ⎟
⎝ 2⎠
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3.
Solve the equation
f (x) = 1
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4.
Calculate the gradient to the curve at x = 3
5.
1
€Find the equation of the tangent line to the curve at x = −
2
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PART V
A. Integrate each of the following with respect to x.
1.
∫
2.
∫
€ 3.
∫
€ 4.
∫
€ 5.
∫
€
6.
∫
€ 7.
∫
€ 8.
3
9 − x2
4
dx
2
dx
2 − 3x
6
dx
9 + x2
2
dx
3+ 4x2
1
4 − (x + 1)
1
2
4 + (x − 3) 2
4
dx
10 − 2x + x 2
−1
1
∫
−5
dx
7 − 6x − x 2
dx
dx
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B. Integrate each of the following:
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1.
2.
€
3.
€
€
2
dx
1− 4 x 2
3x + 4
∫ 2 dx
x +4
2
⎛
2x ⎞
∫ ⎜⎜ 3 + 2 ⎟⎟ dx
⎝
x +4⎠
2 3
4.
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∫
∫
(1+ x) 2
2
2 4+ x
dx
PART VI
A. Find the area of the segment of a circle of radius 12 cm cut off by a chord of length 10
cm.
B. Two circles of radii 15 cm and 8 cm have their centers 17 cm apart. Find the area
common to both circles.
C. A unit circle has a sector cut out of it with a central angle of θ . When a chord is drawn
across the sector with the endpoints each touching the circle, the area of the triangle
formed is 3 times the area of the remaining area of the sector of the circle. Find θ .
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