MATH SOLVE

3 months ago

Q:
# 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

Accepted Solution

A:

Answer:A. The exact value of sec(13π/6) = 2√3/3B. The exact value of cot(7π/4) = -1Step-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 problemA. 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