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
