What Are the Characteristics of This Sine Curve Equation?

[silentthoughts]

Help me with sine questions.. URGENT

Hi I am trying to solve these and i can't understand them .. please help me

Q1:: Write an equation for a sine curve that has the following characteristics: - amplitude of 5, -max of 7, min of -3 and one cycle starts at pi/8 and this cycle ends at 9 pi/8



Q2:the number of hrs of daylight for a particular area is related to the day of the year as follows: D=12 +2.5 sin[2pi/365(t-81)], where D is the number of hours of daylight and t is the day of the year, with t= 1 representing january 1st. find the amplitude, period , phase shift and vertical shift for the equation.

 

[HallsofIvy]

Any "sin curve" can be written y= A sin(bx+ c)+ D. Since the largest possible value of sine is 1 and the smallest is -1, that will have a largest value of A+ D and smallest of -A+ D. One cycle of sin(x) starts at x= 0 and ends at x= 2$\pi$. One cycle of sin(bx+ c), then, starts when bx+ c= 0 and ends when bx+ c= 2$\pi$. Use the information given to find A, b, c, and D.

 

[tacman]

Sure, I'd be happy to help you with these sine questions. Let's start with the first one:

Q1: To write an equation for a sine curve, we can use the general form of y = A sin(Bx + C) + D, where A is the amplitude, B is the frequency, C is the phase shift, and D is the vertical shift.

In this case, we are given the amplitude of 5, which means that A = 5. The maximum value is 7, so we can set D = 7. The minimum value is -3, so we can set D = -3.

Now, for the cycle starting at pi/8 and ending at 9pi/8, we can use the fact that one full cycle of a sine curve is 2pi. This means that the frequency, or B, is equal to 2pi divided by the length of the cycle, which is 9pi/8 - pi/8 = 8pi/8 = pi. So, B = pi.

Putting all of this together, we have the equation y = 5sin(pi x + C) + D. To find the phase shift, we can use the starting point of pi/8. This means that C = -pi/8. Substituting this into our equation, we have y = 5sin(pi x - pi/8) + D.

Q2: For this question, we are given the equation D = 12 + 2.5sin[2pi/365(t-81)]. We can see that the amplitude is 2.5, the period is 365, the phase shift is 81, and the vertical shift is 12.

To find the amplitude, we can use the formula A = |B|, since the coefficient of sine is the amplitude. In this case, A = |2.5| = 2.5.

The period is given as 365, which means that B = 2pi/365.

The phase shift is given as 81, so C = -81.

The vertical shift is given as 12, so D = 12.

Putting all of this together, we have the equation D = 2.5sin[2pi/365(t+81)] + 12.

I hope this helps! Let me know if you have any other questions or