Unit 1
Find the range and domain of the following:
-5 / 8 - 2x - x2
D: (-∞, -4) U (-4, 2) U (2, ∞)
Find the following values:
a) cos(17π)
b) sec(19π / 3)
c) csc(585˚)
a) -1
b) 2
c) -√(2)
Find the following values:
a) cos(15°)
b) tan(12°) + tan(28°) / 1 - tan(12°)tan(28°)
c) cos(7π/12)
a) √(6)/4 +√(2)/4
b) tan(40°)
c) √(2)/4 - √(6)/4
Expand the logarithm:
log√(x2 + 4 / (x2 + 1)(x3 - 7)2)
1/2[log(x2 + 4) - log(x2 + 1) - 2log(x3 - 7)]
sin(θ) = √(3)/2
θ = π/3 + 2πk, 2π/3 + 2πk
Graph the piecewise function:
5, x < -2
1 + x, -2 ≤ x < 1
x2, x ≥ 1
*SEE PICTURE IN WORD DOC*
Given the value of one of the trigonometric functions find the values of the remaining 5 trig functions:
tan(θ) = -(1/7), csc(θ) < 0
sin(θ) = -√(2) / 10
csc(θ) = -5√(2)
cos(θ) = 7√(2) / 10
sec(θ) = 5√(2) = 7
cot(θ) = -7
Find the values of the following:
a) sin(23π/12)
b) cos(-33π/8)
c) tan(31π/12)
a) -√(2 - √(3)) / 2
b) √(2 - √(2)) / 2
c) -2 - √(3)
Rewrite as a single logarithm with coefficient 1:
1/3[log2(x + 1) + log2(x - 1) - 2log2(y)- 1/5(log2(y + 1))]
log3√(x2 - 1 / y2 5√(y + 1))
sin2(θ) = 2sin(θ) + 3
θ = 3π/2 + 2πk
Find the functions f º ɡ, ɡ º f, f º f, and ɡ º ɡ
f(x) = x / x + 1
g(x) = 2x - 1
(f º g)(x) = 2x - 1 / 2x
(g º f)(x) = x - 1 / x + 1
(f º f)(x) = x / 2x + 1
(g º g)(x) = 4x - 3
Find the following values:
a) cos(cos-1(2))
b) tan-1(tan(16π/3))
c) cos(sin-1(-√2/2))
a) undefined
b) π/3
c) √(2)/2
Prove:
sin2θ(1 + cot2θ) = 1
LHS: sin2θ(1 + cos2θ/sin2θ) = 1
sin2θ + sin2θcos2θ/sin2 = 1
You save $1000 from a summer job and put it in a CD earning 5% compounding continuously.
a) How many years will it take for your money to double?
b) What will be your balance after 10 years?
a) t = ln(2) / 0.05 years
b) B = $ 1000e0.5
4cos2(θ) − 4 cos(θ) + 1 = 0
θ = π/3 + 2πk, 5π/3 + 2πk
Graph using D, R, x-int, y-int, HA, & VA:
f(x) = 25 - 53 - x
g(x) = log3(1 - x) - 2
*SEE WORD DOCUMENT*
A water tower is located 313ft from a building. From a window in the building, an observer notes that the angle of elevation to the top of the tower is 43˚ and that the angle of depression to the bottom of the tower is 28˚. How tall is the tower?
a) 313tan(43˚) + 313tan(28˚)
A communications tower is located at the top of a steep hill, as shown. The angle of inclination of the hill is 58°. A guy wire is to be attached to the top of the tower and to the ground, 175 m downhill from the base of the tower. The angle α in the figure is determined to be 12°. Find the length of cable required for the guy wire.
x = 175sin(148°) / sin(20°)
A culture starts with 10,000 bacteria at 10 am, and the number doubles every two hours.
a) Find the function 𝑵(𝒕) = 𝑵0 𝒆rt that models the number of bacteria after t hours.
b) Find the function 𝑵(𝒕) = 𝑵02t/h that models the number of bacteria after t hours.
c) How much bacteria will be present after 25 hours?
d) At what time will the count increase to 24000 bacteria?
a) N(t) = 10000e(ln(2)/2) * t
b) N(t) = 10000(2t/2)
c) N(25) = 10000(225/2)
d) t = 2ln(2.4) / ln(2)
sin(2θ) + cos(θ) = 0
θ = π/2 + πk, 7π/6 + 2πk, 11π/6 + 2πk
Solve:
4x - 1 = 1 / 82x - 3
x = 11/8
Graph for one period, and give the following: Amplitude, Period, Phase shift, x-intercepts, Vertical Asymptotes
f(x) = 3cos(2x + π/2)
A: 3
P: π
Ph. Sh.: π/4 left
x-int: (0, 0), (π/2, 0)
*SEE WORD DOCUMENT FOR GRAPH*
Given tan(θ) = -1/5, 3π/2 < θ < 2π
Find:
a) sin(2θ)
b) cos(2θ)
c) sin(θ/2)
d) cos(θ/2)
e) tan(θ/2)
a) -5/13
b) 12/13
c) √(26 - 5√(21)) / 2√(13)
d) -√(26 + 5√(26)) / 2√(13)
e) 5 - √(26)
Graph for one period, and give the following: Amplitude, Period, Phase shift, x - intercepts, Vertical Asymptotes
𝑓(𝑥) = −sec(3𝑥 + 𝜋/2 )
A: None
P: 2π/3
Ph. Sh.: π/6 left
x - int: None
VA: x = 0, x = π/3, x = 2π/3
*SEE WORD DOCUMENT FOR GRAPH*
2 sin(θ) tan(θ) − tan(θ) = 1 − 2 sin(θ)
θ = π/6 + 2πk, 5π/6 + 2πk, 3π/4 + πk