# Answer to Question #6026 in Calculus for Samantha

Question #6026

Evaluate the trigonometric function for each value of theta

theta=(-3pi/4), find sin(theta),cos(theta),tan(theta),cot(theta),sec(theta),csc(theta).

theta=(-pi/4), find sin(theta),cos(theta),tan(theta),cot(theta),sec(theta),csc(theta).

theta=(2pi/3), find sin(theta),cos(theta),tan(theta),cot(theta),sec(theta),csc(theta).

theta=(-3pi/4), find sin(theta),cos(theta),tan(theta),cot(theta),sec(theta),csc(theta).

theta=(-pi/4), find sin(theta),cos(theta),tan(theta),cot(theta),sec(theta),csc(theta).

theta=(2pi/3), find sin(theta),cos(theta),tan(theta),cot(theta),sec(theta),csc(theta).

Expert's answer

Evaluate the trigonometric function for each value of α

α=(-3pi/4), find sin(α),cos(α),tan(α),cot(α),sec(α),csc(α).

α=(-pi/4), find sin(α),cos(α),tan(α),cot(α),sec(α),csc(α).

α=(2pi/3), find sin(α),cos(α),tan(α),cot(α),sec(α),csc(α).

We'll use next expressions:

tan(α) = sin(α)/cos(α)

cot(α) = 1/tan(α)

sec(α) = 1/cos(α)

csc(α) = 1/sin(α)

α=(-3pi/4):

sin(-3pi/4) = -1/sqrt(2)

cos(-3pi/4) = -1/sqrt(2)

tan(-3pi/4) = (-1/sqrt(2))/(-1/sqrt(2)) = 1

cot(-3pi/4) = 1/1 = 1

sec(-3pi/4) = 1/(-1/sqrt(2)) = -sqrt(2)

csc(-3pi/4) = 1/(-1/sqrt(2)) = -sqrt(2)

α=(-pi/4):

sin(-pi/4) = -1/sqrt(2)

cos(-pi/4) = 1/sqrt(2)

tan(-pi/4) = (-1/sqrt(2))/(1/sqrt(2)) = -1

cot(-pi/4) = (1/sqrt(2))/(-1/sqrt(2)) = -1

sec(-pi/4) = 1/(1/sqrt(2)) = sqrt(2)

csc(-pi/4) = 1/(-1/sqrt(2)) = -sqrt(2)

α=(2pi/3):

sin(2pi/3) = sqrt(3)/2

cos(2pi/3) = -1/2

tan(2pi/3) = (sqrt(3)/2)/(-1/2) = -sqrt(3)

cot(2pi/3) = -1/sqrt(3)

sec(2pi/3) = -2

csc(2pi/3) = 2/sqrt(3)

α=(-3pi/4), find sin(α),cos(α),tan(α),cot(α),sec(α),csc(α).

α=(-pi/4), find sin(α),cos(α),tan(α),cot(α),sec(α),csc(α).

α=(2pi/3), find sin(α),cos(α),tan(α),cot(α),sec(α),csc(α).

We'll use next expressions:

tan(α) = sin(α)/cos(α)

cot(α) = 1/tan(α)

sec(α) = 1/cos(α)

csc(α) = 1/sin(α)

α=(-3pi/4):

sin(-3pi/4) = -1/sqrt(2)

cos(-3pi/4) = -1/sqrt(2)

tan(-3pi/4) = (-1/sqrt(2))/(-1/sqrt(2)) = 1

cot(-3pi/4) = 1/1 = 1

sec(-3pi/4) = 1/(-1/sqrt(2)) = -sqrt(2)

csc(-3pi/4) = 1/(-1/sqrt(2)) = -sqrt(2)

α=(-pi/4):

sin(-pi/4) = -1/sqrt(2)

cos(-pi/4) = 1/sqrt(2)

tan(-pi/4) = (-1/sqrt(2))/(1/sqrt(2)) = -1

cot(-pi/4) = (1/sqrt(2))/(-1/sqrt(2)) = -1

sec(-pi/4) = 1/(1/sqrt(2)) = sqrt(2)

csc(-pi/4) = 1/(-1/sqrt(2)) = -sqrt(2)

α=(2pi/3):

sin(2pi/3) = sqrt(3)/2

cos(2pi/3) = -1/2

tan(2pi/3) = (sqrt(3)/2)/(-1/2) = -sqrt(3)

cot(2pi/3) = -1/sqrt(3)

sec(2pi/3) = -2

csc(2pi/3) = 2/sqrt(3)

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