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

sin(x) has amplitude 1, period 2pi, no phase shift

sin(x-pi/3) is shifted pi/3 to the right
sin(2(x-pi/3)) has period pi, shifted right by pi/3
2sin(2(x-pi/3)) meets the requirements.

Do the others in lime wise.
Recall that tan(x) has period pi, not 2pi.

To construct the equations of the given trigonometric functions, we need to consider the properties provided for each function and understand how they affect the standard equations. The standard equations for sine, cosine, and tangent are as follows:

Sine function: y = A * sin(B(x - C)) + D
Cosine function: y = A * cos(B(x - C)) + D
Tangent function: y = A * tan(B(x - C)) + D

Let's go through each part of the question to determine the equation for each function.

A) Sine function with amplitude 2, period π, phase shift π/3 to the right:
Amplitude: The amplitude refers to the vertical stretch or compression. In this case, the amplitude is 2.
Period: The standard period of the sine function is 2π. However, since the period is given as π, it means the function is compressed horizontally by a factor of 1/2.
Phase shift: The phase shift determines the horizontal translation of the function. Given a phase shift of π/3 to the right, we need to substitute C as π/3 in the equation.
Equation: y = 2 * sin(x - π/3)

B) Tangent function with reflection in the y-axis, period 3/4, translation up 5 units:
Reflection: Reflection in the y-axis changes the sign of the x in the standard equation. Therefore, we use -B in the equation.
Period: The standard period of the tangent function is π. However, since the period is given as 3/4, it means the function is compressed horizontally by a factor of 4/3.
Translation: The translation up 5 units implies a vertical shift of 5 units, so we substitute D as 5 in the equation.
Equation: y = -tan((4/3)x) + 5

C) Cosine function with period 270°, translation down 50 units, reflection in the x-axis:
Period: The standard period of the cosine function is 2π. Since the period is given as 270°, we need to convert it to radians (π) and then compress it horizontally by a factor of 2/3.
Translation: The translation down 50 units implies a vertical shift of -50 units, so we substitute D as -50 in the equation.
Reflection: Reflection in the x-axis changes the sign of the entire function, so we use -A in the equation.
Equation: y = -2 * cos((2/3)x) - 50

By considering the given properties and applying them to the standard equations of the trigonometric functions, we can construct the equations as shown above for each function.