Download Pre-calculus Final Review Study Guide

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Pre-calculus Final Review Study Guide
This study guide was made by Nicole Scavarda, class of 2010.
Chapter 7: Trigonometric Identities and Equations
(7.1) p. 427
Use the given information to determine the exact trigonometric value.
9.) cot ө = - √5 / 2 , π/2 < ө < π; tan ө
11.) tan ө = -4/7, 270˚< ө < 360˚; sec ө
Simplify each expression.
14.) csc ө
15.) cos x csc x tan x
16.) cos x cot x + sin x
cot ө
(7.2) p. 434
Verify that each equation is an identity.
5.) cos x = cot x
7.) csc ө – cot ө = 1 / csc ө + cot ө
csc x
13.) tan A = sec A
csc A
15.) sec x – tan x = 1- sin x
cos x
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(7.3) p. 442
Use sum or difference identities to find the exact value of each trigonometric function.
14.) cos 105
˚
17.) sin π/12
Find each exact value if 0 < x < π /2 and 0 < y < π /2.
27.) cos (x – y) if cos x = 3/5 and cos y = 4/5
28.) tan (x – y) if sin x = 8/17 and cos y = 3/5
Verify that each equation is an identity.
37.) cos (180 ˚+ x) = - cos x
(7.4) p. 453
Use the given information to find sin 2ө, cos 2ө, and tan 2ө.
9.) tan ө = 4/3, π < ө < 3π/2
21.) cos ө = 4/5, 0˚ < ө < 90˚
23.) tan ө = -2, π/2 < ө < π
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Use the half-angle identity to find the exact value of each function
15.) sin 75
˚
(7.5) p. 459
Solve each equation for principal values of x. Express solution in degrees.
17.) √2 sin x – 1 = 0
18.) 2 cos x + 1 = 0
21.) cos^2 x = cos x
Solve each equation for 0˚< x < 360˚.
25.) sin x tan x – sin x = 0
Solve each equation for all real values of x.
39.) 3 tan^2 x = √3 tan x
46.) sec^2 x + 2 sec x = 0
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Chapter 9: Polar Coordinates
(9.1) p. 558
Graph each point.
6.) A(1, 135˚)
7.) B(2.5, -π/6)
Find the distance between the points with the given polar coordinates.
42.) P1 (4, 170˚) and P2 (6, 105˚)
46.) Find an ordered pair of polar coordinates to represent the point whose
rectangular coordinates are (-3, 4).
(9.3) p. 571
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Find the polar coordinates of each point with the given rectangular coordinates. Use 0 < ө < 2π
and r > 0
5.) (- √2, √2 )
Write each rectangular equation in polar form.
9.) y = 2
10.) x^2 +y^2 = 16
26.) x = -7
Write each polar equation in rectangular form.
11.) r = 6
Find the rectangular coordinates of each point with the given polar coordinates.
23.) (-2, 270˚)
20.) (3, π/2)
(9.5) p. 583
Simplify.
5.) i^-6
6.) i^10 + i^2
8.) (2.3 + 4.1i) – (-1.2 – 6.3i)
10.) (-2 - i) ^2
7.) (2+3i) + (-6+ i)
9.) (2+4 i) + (-1 + 5i)
11.) i / (1+ 2i)
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13.) i^6
15.) i^1776
19.) (1/2 + i) – (2 - i)
17.) (3 + 2i) + (-4 + 6i)
21.) (2+ i)(4 + 3i)
23.) (1 + √7 i)(-2 - √5 i)
25.) (2+ i) / (1+ 2i)
(9.6) p. 589
5.) Solve the equation 2x + y + x i + y i = 5 + 4 i for x and y, where x and y are real
numbers.
Graph each number in the complex plane and find its absolute value.
6.) -2 – i
7.) 1+ √2 i
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Express each complex number in polar form.
8.) 2 - 2 i
9.) 4 + 5 i
10.) -2
Graph each complex number. Then express it in rectangular form.
11.) 4(cos π/3 + i sin π/3)
12.) 2(cos 3 + i sin 3)
13.) 3/2 (cos 2π + i sin 2π)
Solve the equation for x and y, where x and y are real numbers.
17.) 1 + (x+y) i = y + 3xi
Express each complex number in polar form.
29.) -4 + i
33.) -4 √2
Graph each number in the complex plane and find its absolute value.
21.) -1 - 5i
25.) Find the modulus of z = -4 + 6i.
Graph each complex number. Then express it in rectangular form.
37.) 2(cos 4π/3 + i sin 4π/3)
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(9.7) p. 596
Find each product or quotient. Express the result in rectangular form.
5.) 3(cos π/6 + i sin π/6) / 4(cos 2π/3 + i sin 2π/3)
7.) ½ (cos π/3 + i sin π/3) * 6(cos 5π/6 + i sin 5π/6)
11.) 6(cos 3π/4 + i sin 3π/4) / 2(cos π/4 + i sin π/4)
13.) 5(cos π + i sin π) * 2(cos 3π/4 + i sin 3π/4)
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Chapter 11: Exponential and Logarithmic Functions
(11.1) p. 700
Evaluate the expression.
4.) 5^ -4
5.) (9/16)^ -2
6.) 216^(1/3)
Simplify each expression.
9.) (3a^-2)^3 * 3a^5
10.) √(m^3n^2) * √(m^4n^5
12.) (2x^4y^8)^(1/2)
Express using rational exponents.
13.) √ (169x^5)
14.)
4
√ (a^2b^3c^4d^5)
Express using a radical.
15.) 6^(1/4)b^(3/4)c^(1/4)
(11.2) p. 708
8.) Business owners keep track of the value of their assets for tax purposes. Suppose the value of
a computer depreciates at a rate of 25% a year. Determine the value of the laptop computer two
years after it has been purchased for $3750.
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Graph each exponential function or inequality.
10.) y = 2^x
11.) y = -2^x
12.) y = 2^ -x
13.) y =2^(x + 3)
(11.3) p.714
6.) Bakersfield, California was founded in 1859 when Colonel Thomas Baker planted ten acres
of alfalfa for travelers going from Visalia to Los Angeles to feed their animals. The city’s
population can be modeled by the equation y = 33,430e ^(0.0397t), where t is the number of
years since 1950.
a.) Has Bakersfield experienced growth or decline in population?
b.) What was Bakersfield’s population in 1950?
c.) Find the projected population of Bakersfield in 2010.
7.) The Kwans are saving for their daughter’s college education. If they deposit $12,000 in an
account bearing 6.4% interest compounded continuously, how much will be in the account when
Ann goes to college in 12 years?
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(11.4) p.722
Write each equation in exponential form.
6.) log9 27 = 3/2
7.) log(1/25) 5 = -1/2
Write each equation in logarithmic form.
8.) 7^ -6 = y
9.) 8^ (-2/3) = 1/4
Evaluate each expression.
10.) log2 (1/16)
11.) log10 0.01
12.) log7 (1/343)
Solve each equation.
13.) log2 x = 5
14.) log7 n = (2/3)log 7 8
15.) log6 (4x + 4) = log 6 64
16.) 2 log6 4 – (1/4) log6 16 = log6 x
(11.5) p.730
Find the value of each logarithm using the change of base formula.
11.) log12 18
Solve each equation or inequality.
40.) 2^x = 95
12.) log8 15