AMA OED
PRE-CALCULUS
WEEK 11-20
Plot the point given in polar coordinates and find two additional polar representations of the
point, using
-2π < θ < 2π.
(4, - π/3)
Answer: (4, 5 π/3), (-4, -4 π/3)
Determine all solutions of each equation in radians (for x) or degrees (for θ) to the
nearest tenth as appropriate.
2cos^2+cosx=1
Answer: π/3 + 2nπ, π + 2nπ, 5π/3 + 2nπ, where n is any integer
Solve the equation for exact solutions over the interval [0, 2π].
2√3sin2x=√3
Answer: {π/12,5π/12,13π/12,17π/12}
Solve each equation for exact solutions over the interval [00, 3600].
(cotθ−√3)(2sinθ+√3)=0
Answer: {300, 2100, 2400, 3000}
Find a polar equation of the conic with its focus at the pole.
Conic: Parabola, Vertex or vertices: (1, -π/2)
Answer: 2/1−sinθ
Find the exact value of the tangent of the angle by using a sum or difference formula.
-165°
Answer: tan (-165)° = 2 – √3
Convert the polar equation to rectangular form.
r^2 = cosθ
Answer: X2 + y2 – x2/3 = 0
Give all exact solutions over the interval [00, 3600].
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sin2θ=2cos^2θ
Answer: 45° + 360°n, 90° + 360°n, 225° + 360°n, 270° + 360°n, where n is any
integer
Convert the polar equation to rectangular form.
r = 4sinθ
Answer: x^2 + y^2 – 4y = 0
Plot the point given in polar coordinates and find two additional polar representations of
the point, using -2π < θ < 2π.
(0,−7π60,−7π6)
Answer: (0, 5π/6), (0, -13π/6)
Convert the polar equation to rectangular form.
r=4
Answer: x^2+y^2 = 16
Solve the equation for exact solutions over the interval [0, 2π].
sinx/2=√2−sinx/2
Answer: {π/2,3π/12}
Convert the polar equation to rectangular form.
r^2 = cos θ
Answer: X2 + y2 – x2/3 = 0
Convert the polar equation to rectangular form.
r = 6/2-3sin θ
Answer: 4x2 – 5y2 – 36y – 36 = 0
Convert the polar equation to rectangular form.
r=2sin3θ
Answer: (x2 + y2)2 = 6x2y – 2y3
Convert the polar equation to rectangular form.
θ=2π/3
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Answer: √3x + y = 0
Convert the polar equation to rectangular form.
r=4cscθ
Answer: y = 4
Write the expression as the sine, cosine, or tangent of an angle.
tan2x+tanx/1−tan2xtanx
Answer: tan3x=sin(3x)/cos(3x)
Convert the polar equation to rectangular form.
r = 2/1 + sin θ
Answer: X + 4y – 4 = 0
2
Find a polar equation of the conic with its focus at the pole.
Conic: Hyperbola, Eccentricity: e = 2, Directrix: x = 1
Answer: r = 2/1+2cos θ
Convert the rectangular equation to polar form. Assume a > 0.
xy = 16
Answer: r^2 = 16sec θcsc θ = 32csc2 θ
Convert the rectangular equation to polar form. Assume a > 0.
3x - y + 2 = 0
Answer: r= −2/3cosθ−sinθ
A point in polar coordinates is given. Convert the point to rectangular coordinates.
(3, π/2)
Answer: (0, 3)
Convert the rectangular equation to polar form. Assume a > 0.
x2 + y2 - 2ax = 0
Answer: r = 2acosθ
Solve the equation for exact solutions over the interval [0, 2π].
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sin3x=−1
Answer: {π/2,7π/6,11π/6}
Plot the point given in polar coordinates and find two additional polar representations of
the point, using -2π < θ < 2π.
(√2,2.36)
Answer: (√2, 8.64), (-√2, -0.78)
Convert the rectangular equation to polar form. Assume a > 0.
y2 - 8x - 16 = 0
Answer: r = 4/1−cosθ or −4/1+cosθ
Solve the equation for exact solutions over the interval [0, 2π].
tan 4x = 0
Answer: {0, π/4, π/2, 3π/4, π, 5π/4, 3π/2, 7π 4}
Determine all solutions of each equation in radians (for x) or degrees (for θ) to the
nearest tenth as appropriate.
3sin^2x−sinx−1=0
Answer: 9 + 2nπ, 2.3 + 2nπ, 3.6 + 2nπ, 5.8 + 2nπ, where n is any integer
Find a polar equation of the conic with its focus at the pole.
Conic: Ellipse, Eccentricity: e = 1/2, Directrix: x = 1
Answer: r = 1/2+sinθ
Plot the point given in polar coordinates and find two additional polar representations of
the point, using -2π < θ < 2π.
(2√2, 4.71)
Answer: (2√2, 10.99), (-2√2, 7.85)
Write the expression as the sine, cosine, or tangent of an angle.
sin 3 cos 1.2 - cos 3 sin 1.2
Answer: sin 1.8
Find a polar equation of the conic with its focus at the pole.
Conic: Parabola, Eccentricity: e = 1, Directrix: x = -1
Answer: r = 1/1-cos θ
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Find the exact value of the tangent of the angle by using a sum or difference formula.
-165°
Answer: tan (-165)° = −2√4(3–√+1
Find a polar equation of the conic with its focus at the pole.
Conic: Parabola, Vertex or vertices: (5, π)
Answer: r = 10/1cosθ
A point in polar coordinates is given. Convert the point to rectangular coordinates.
(-1, 5π/4)
Answer: (√2/2, √2,2)
Find the exact value of each expression.
a. cos (120° + 45°)
b. cos120° + cos45°
Answer: (a) −√2−√6/4 (b) −1+√2/2
Write the first five terms of the sequence. Assume that n begins with 1.
An = n(n - 1)(n - 2)
Answer: 0, 0, 6, 24, 60
Solve each equation for exact solutions over the interval [00, 3600].
(tanθ−1)(cosθ−1)=0
Answer: {00, 450, 2250}
Write the expression as the sine, cosine, or tangent of an angle.
cos 25° cos 15° - sin 25° sin 15°
Answer: cos 40
Solve the equation for exact solutions over the interval [0, 2π].
sin 3x = 0
Answer: {0,π/3,2π/3,π,4π/3,5π/3}
Give all exact solutions over the interval [0°, 360°].
2−sin2θ=4sin2θ
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Answer: 11.8° + 360°n, 78.2° + 360°n, 191.8° + 360°n, 258.2° + 360°n, where n is any
integer.
Give all exact solutions over the interval [0°, 360°].
2cos^22θ=1−cos2θ
Answer: 30° + 360°n, 90° + 360°n, 150° + 360°n, 210° + 360°n, 270° + 360°n, 330° +
360°n, where n is any integer.
Give all exact solutions over the interval [00, 3600].
sinθ−sin2θ=0
Answer: 0° + 360°n, 60° + 360°n, 180° + 360°, 300° + 360°n, where n is any integer.
Give all exact solutions over the interval [0°, 360°].
4cos2θ=8sinθcosθ
Answer: 22.5° + 360°n, 112.5° + 360°n, 202.5° + 360°n, 292.5° + 360°n, where n is
any integer
Solve the equation for exact solutions over the interval [0, 2π].
√2cos2x=−1
Answer: {3π/8,5π/8,11π/18,13π/18}
Solve each equation for exact solutions over the interval [00, 3600].
2sinθ−1=cscθ
Answer: {900, 2100, 3300}
Convert the rectangular equation to polar form. Assume a > 0.
y=4
Answer: R = 4 csc θ
Solve the equation for exact solutions over the interval [0, 2π].
3tan3x = √3
Answer: {π/18,7π/18,13π/18,19π/18,25π/18,31π/18}
Give all exact solutions over the interval [0°, 360°].
2cos22θ=1−cos2θ
Answer: 30° + 360°n, 90° + 360°n, 150° + 360°n, 210° + 360°n, 270° + 360°n, 330° +
360°n, where n is any integer.
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Give all exact solutions over the interval [00, 3600].
csc2θ2=2secθ
Answer: 60° + 360°n, 300° + 360°n, where n is any integer.
Solve the equation for exact solutions over the interval [0, 2π].
cot3x=√3
Answer: {π/17,7π/17,13π/17,19π/17,25π/17,31π/17}
Give all exact solutions over the interval [00, 3600].
cosθ=sin^2θ/2
Answer: 70.5° + 360°n, 289.5° + 360°n, where n is any integer
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