Construct the equations of the following trigonometric functions:

A)A sine function with amplitude 2, period , phase shift /3 right

B)A tangent function with a reflection in the y-axis, period ¾, translation up 5 units

C)A cosine function with period 270°, translation down 50 units, reflection in the x-axis

To construct the equations of the trigonometric functions with the given conditions, we need to understand the general forms of these functions and how each condition affects them.

A) For a sine function, the general form is: y = A sin(Bx + C) + D

Amplitude: The amplitude determines the peak value of the function. In this case, the given amplitude is 2, so A = 2.

Period: The period determines the distance between consecutive peaks of the function. The given period is π, which is the standard period for the sine function. In this case, B = 2π/π = 2.

Phase Shift: The phase shift determines the horizontal displacement of the function. Since the phase shift is π/3 to the right, the value inside the parentheses becomes (x - π/3). Therefore, C = -π/3.

Putting all the values together, the equation of the sine function is: y = 2 sin(2x - π/3).

B) For a tangent function, the general form is: y = A tan(Bx + C) + D

Reflection in the y-axis: A reflection in the y-axis changes the sign of the equation's coefficient A. Since the given reflection exists, A will have a negative value.

Period: The period determines the distance between consecutive peaks/valleys of the function. The given period is 3/4, so B = 2π/(3/4) = (8π)/3.

Translation up: A translation up shifts the function vertically. The given translation is up 5 units, so D = 5.

Putting all the values together, the equation of the tangent function is: y = -tan((8π/3)x) + 5.

C) For a cosine function, the general form is: y = A cos(Bx + C) + D

Period: The period determines the distance between consecutive peaks of the function. The given period is 270°, which is equivalent to (3/4)π. In this case, B = 2π/(3/4π) = (8π)/3.

Translation down: A translation down shifts the function vertically. The given translation is down 50 units, so D = -50.

Reflection in the x-axis: A reflection in the x-axis changes the sign of the coefficient A. Since there is a reflection in the x-axis, A will have a negative value.

Putting all the values together, the equation of the cosine function is: y = -cos((8π/3)x) - 50.

These are the equations for the trigonometric functions in the given scenarios.

A) The general equation of a sine function is given by y = A*sin(B(x-C)), where A represents the amplitude, B represents the frequency (or 1/period), and C represents the phase shift.

For this problem, the amplitude is given as 2, the period is not stated, but we can calculate it using the formula period = 2π/B, and the phase shift is /3 right.

Knowing that the formula for period is 2π/B, we can rearrange the formula to solve for B.
Period = 2π/B
B = 2π/Period

Substituting the given period = 2π, we get B = 2π/(2π) = 1.

Therefore, the equation for the sine function is y = 2*sin(x - /3).

B) The general equation of a tangent function is given by y = A*tan(B(x-C)), where A represents the amplitude, B represents the frequency (or 1/period), and C represents the phase shift.

For this problem, the function has a reflection in the y-axis, a period of ¾, and a translation up 5 units.

Reflecting a function in the y-axis simply changes the sign of the amplitude. Therefore, the amplitude A becomes -A.

The given period is ¾, which means the frequency B is equal to 1/period = 1/(3/4) = 4/3.

The translation up 5 units means we add 5 to the function.

Therefore, the equation for the tangent function is y = -A*tan(B(x-C)) + D.
Substituting the given values, we get y = -tan((4/3)(x-C)) + 5.

C) The general equation of a cosine function is given by y = A*cos(B(x-C)), where A represents the amplitude, B represents the frequency (or 1/period), and C represents the phase shift.

For this problem, the function has a period of 270°, a translation down 50 units, and a reflection in the x-axis.

The given period is 270°, which means the frequency B is equal to 1/period = 1/(270/360) = 4/3.

The translation down 50 units means we subtract 50 from the function.

A reflection in the x-axis changes the sign of the function. Therefore, the amplitude A becomes -A.

Therefore, the equation for the cosine function is y = -A*cos(B(x-C)) + D.
Substituting the given values, we get y = -cos((4/3)(x-C)) - 50.