A. in two or more complete sentences, explain how to find the exact value of sec 13pi/6 including quadrant location

b. in two or more complete sentences, explain how to find the exact value of cot 7pi/4 including quadrant location

A. The exact value of sec(13π/6) = 2√3/3

B. The exact value of cot(7π/4) = -1

Step-by-step explanation:

* Lets study the four quadrants

# First quadrant the measure of all angles is between 0 and π/2

  the measure of any angle is α  

∴ All the angles are acute  

∴ All the trigonometry functions of α are positive

# Second quadrant the measure of all angles is between π/2 and π

  the measure of any angle is π - α

∴ All the angles are obtuse

∴ The value of sin(π - α) only is positive

  sin(π - α) = sin(α)  ⇒ csc(π - α) = cscα

  cos(π - α) = -cos(α)   ⇒ sec(π - α) = -sec(α)

  tan(π - α) = -tan(α)   ⇒ cot(π - α) = -cot(α)

# Third quadrant the measure of all angles is between π and 3π/2

  the measure of any angle is π + α  

∴ All the angles are reflex  

∴ The value of tan(π + α) only is positive

  sin(π + α) = -sin(α)  ⇒ csc(π + α) = -cscα

  cos(π + α) = -cos(α)   ⇒ sec(π + α) = -sec(α)

  tan(π + α) = tan(α)   ⇒ cot(π + α) = cot(α)

# Fourth quadrant the measure of all angles is between 3π/2 and 2π  

  the measure of any angle is 2π - α  

∴ All the angles are reflex

∴ The value of cos(2π - α) only is positive

  sin(2π - α) = -sin(α)  ⇒ csc(2π - α) = -cscα

  cos(2π - α) = cos(α)   ⇒ sec(2π - α) = sec(α)

  tan(2π - α) = -tan(α)   ⇒ cot(2π - α) = -cot(α)

* Now lets solve the problem

A. The measure of the angle 13π/6 = π/6 + 2π

- The means the terminal of the angle made a complete turn (2π) + π/6

∴ The angle of measure 13π/6 lies in the first quadrant

∴ sec(13π/6) = sec(π/6)

∵ sec(x) = 1/cos(x)

∵ cos(π/6) = √3/2

∴ sec(π/6) = 2/√3 ⇒ multiply up and down by √3

∴ sec(π/6) = 2/√3 × √3/√3 = 2√3/3

* The exact value of sec(13π/6) = 2√3/3

B. The measure of the angle 7π/4 = 2π - π/4

- The means the terminal of the angle lies in the fourth quadrant

∴ The angle of measure 7π/4 lies in the fourth quadrant

- In the fourth quadrant cos only is positive

∴ cot(2π - α) = -cot(α)

∴ cot(7π/4) = -cot(π/4)

∵ cot(x) = 1/tan(x)

∵ tan(π/4) = 1

∴ cot(π/4) = 1

∴ cot(7π/4) = -1

* The exact value of cot(7π/4) = -1

a, b, c, d are correct.

step-by-step explanation:

we have g(x) = 3 and are required to find which option is a translation of g(x).

now, 'translation' is a transformation which means to shift the function in any direction and the general form of translation is g(x) + k, where k is any real number.

so, according to the options we see that all are correct as they satisfy the general form of translation.

also, it can be seen from the graph below that the function g(x) = 3 is shifted from its position in all the four options.