Question

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

Answer 1

Answer:

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