Verify these answers
1.Which of the six trigonometric functions are positive in quadrant III?
Answer: tangent and cotangent
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2. What is the amplitude of the function y = pi/2 sin(6/3t - pi/6) + 14?
Amplitude: pi/2
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3. What is the function of cos that has an amplitude of 4, a period of 2pi and has a point at (0,2)?
Answer: y=4cos(x-0) + 2
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4. The height of a swing is modelled by the function h(t) 35sin(8pi x t)+74. What are the highest and lowest points the ball reaches where h(t) is in cm?
when x=0, cos(0) = 1
so you want
y = 4cos(x) - 2
74 ± 35
To verify these answers and understand how to arrive at them, let's go through each question step by step:
1. Which of the six trigonometric functions are positive in quadrant III?
To determine the positive trigonometric functions in quadrant III, we can use the acronym "All Students Take Calculus," which stands for:
- All functions are positive in Quadrant I.
- Sine (sin) is positive in Quadrant II.
- Tangent (tan) is positive in Quadrant III.
- Cosine (cos) is positive in Quadrant IV.
Therefore, the correct answer is indeed tangent and cotangent.
2. What is the amplitude of the function y = (π/2)sin((6/3)t - (π/6)) + 14?
The general form of a trigonometric function is y = A*sin(Bx + C) + D, where A represents the amplitude. In this case, A is equal to (π/2). Therefore, the amplitude is π/2.
3. What is the function of cos that has an amplitude of 4, a period of 2π, and a point at (0,2)?
The general form of a cosine function is y = A*cos(Bx - C) + D. Here, we are given that the amplitude is 4, which means |A| = 4. Since cosine functions never exceed an amplitude of 1, the amplitude value should be positive. So we take A = 4.
The given point (0,2) tells us that the midline (vertical shift) is at 2. So, D = 2.
The period of the cosine function is 2π. The formula for the period is T = (2π)/|B|. Given T = 2π, we can solve for |B| = 1.
Hence, the function of cos(x) that satisfies all the given conditions is y = 4*cos(x - 0) + 2, which can simplify to y = 4*cos(x) + 2.
4. The height of a swing is modeled by the function h(t) = 35sin(8πt) + 74. What are the highest and lowest points the swing reaches where h(t) is in cm?
To find the highest and lowest points of the swing, we need to consider the amplitude and vertical shift of the trigonometric function.
The amplitude of h(t) is given by the coefficient of sin, which is 35. Therefore, the maximum and minimum values of the function are ±35.
To find the highest point, we add the amplitude to the vertical shift: 35 + 74 = 109 cm.
To find the lowest point, we subtract the amplitude from the vertical shift: -35 + 74 = 39 cm.
Therefore, the highest point the swing reaches is 109 cm, and the lowest point is 39 cm.