17.3: Exercises
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Find sin(x), cos(x), and tan(x) for the following angles.
- x=120∘
- x=390∘
- x=−150∘
- x=−45∘
- x=1050∘
- x=−810∘
- x=5π4
- x=5π6
- x=10π3
- x=15π2
- x=−π6
- x=−54π8
- Answer
-
- sin(120∘)=√32,cos(120∘)=−12,tan(120∘)=−√3
- sin(390∘)=12,cos(390∘)=√32,tan(390∘)=√33
- sin(−150∘)=−12,cos(−150∘)=−√32,tan(−150∘)=√33
- sin(−45∘)=−√22,cos(−45∘)=√22,tan(−45∘)=−1
- sin(1050∘)=−12,cos(1050∘)=√32,tan(1050∘)=−√33
- sin(−810∘)=−1,cos(−810∘)=0,tan(−810∘) is undefined
- sin(5π4)=−√22,cos(5π4)=−√22,tan(5π4)=1
- sin(5π6)=12,cos(5π6)=−√32,tan(5π6)=−√33
- sin(10π3)=−√32,cos(10π3)=−12,tan(10π3)=√3
- sin(15π2)=−1,cos(15π2)=0,tan(15π2) is undefined
- sin(−π6)=−12,cos(−π6)=√32,tan(−π6)=−√33
- sin(−54π8)=−√22,cos(−54π8)=−√22,tan(−54π8)=1
Graph the function, and describe how the graph can be obtained from one of the basic graphs y=sin(x), y=cos(x), or y=tan(x).
- f(x)=sin(x)+2
- f(x)=cos(x−π)
- f(x)=tan(x)−4
- f(x)=5⋅sin(x)
- f(x)=cos(2⋅x)
- f(x)=sin(x−2)−5
- Answer
-
- shift y=sin(x) up by 2
- y=cos(x) shifted to the right by π
- y=tan(x) shifted down by 4
- y=sin(x) stretched away from the x-axis by a factor 5
- y=cos(x) compressed towards the y-axis by a factor 2
- y=sin(x) shifted to the right by 2 and down by 5
- shift y=sin(x) up by 2
Identify the formulas with the graphs. f(x)=sin(x)+2,g(x)=tan(x−1),h(x)=3sin(x),i(x)=3cos(x),j(x)=cos(x−π),k(x)=tan(x)−1
- Answer
-
- g(x)
- h(x)
- j(x)
- k(x)
- i(x)
- f(x)
Find the formula of a function whose graph is the one displayed below.
- Answer
-
- y=5cos(x)
- y=−5cos(x)
- y=−5sin(x)
- y=cos(x)+5
- y=sin(x)+5
- y=2sin(x)+3
Find the amplitude, period, and phase-shift of the function.
- f(x)=5sin(2x+3)
- f(x)=sin(πx−5)
- f(x)=6sin(4x)
- f(x)=−2cos(x+π4)
- f(x)=8cos(2x−6)
- f(x)=3sin(x4)
- f(x)=−cos(x+2)
- f(x)=7sin(2π5x−6π5)
- f(x)=cos(−2x)
- Answer
-
- amplitude 5, period π, phase-shift −32
- amplitude 1, period 2, phase-shift 5π
- amplitude 6, period π2, phase-shift 0
- amplitude 2, period 2π, phase-shift −π4
- amplitude 8, period π, phase-shift 3
- amplitude 3, period 8π, phase-shift 0
- amplitude 1, period 2π, phase-shift −2
- amplitude 7, period 5, phase-shift 3
- amplitude 1, period π, phase-shift 0
Find the amplitude, period, and phase-shift of the function. Use this information to graph the function over a full period. Label all maxima, minima, and zeros of the function.
- y=5cos(2x)
- y=4sin(πx)
- y=2sin(2π3x)
- y=cos(2x−π)
- y=cos(πx−π)
- y=−6cos(−x4)
- y=−cos(4x+π)
- y=7sin(x+π4)
- y=5cos(x+3π2)
- y=4sin(5x−π)
- y=−3cos(2πx−4)
- y=7sin(14x+π4)
- y=cos(3x−4π)
- y=2sin(15x−π10)
- y=13cos(145x−6π5)
- Answer
-
- amplitude 5, period π, phase-shift 0
- amplitude 4, period 2, phase-shift 0
- amplitude 2, period 3, phase-shift 0
- amplitude 1, period π, phase-shift π2
- amplitude 1, period 2, phase-shift 1
- amplitude 6, period 8π, phase-shift 0
- amplitude 1, period π2, phase-shift −π4
- amplitude 7, period 2π, phase-shift −π4
- amplitude 5, period 2π, phase-shift −3π2
- amplitude 4, period 2π5, phase-shift π5
- amplitude 3, period 1, phase-shift 2π
- amplitude 7, period 8π, phase-shift −π
- amplitude 1, period 2π3, phase-shift 4π3
- amplitude 2, period 10π, phase-shift π2
- amplitude 13, period 5π7, phase-shift 3π7
- amplitude 5, period π, phase-shift 0