Question

Use the given conditions to find the values of all six trigonometric functions. (If answer is undefined, enter UNDEFINED.)

cos(θ)=1/3, sin(θ) < 0
sin(θ)=?
cos(θ)=?
tan(θ)=?
csc(θ)=?
sec(θ)=?
cot(θ)=?

Answer 1

To find the values of the six trigonometric functions, we use the given value of cos(θ) = 1/3 and determine sin(θ) using the Pythagorean identity. Then, we can find the values of tan(θ), csc(θ), sec(θ), and cot(θ) using the formulas.

Step-by-step explanation:

To find the values of the six trigonometric functions, we need to determine the values of sin(θ), cos(θ), tan(θ), csc(θ), sec(θ), and cot(θ) given that cos(θ) = 1/3 and sin(θ) < 0.

Since cos(θ) = 1/3, we can determine sin(θ) using the Pythagorean identity sin²(θ) + cos²(θ) = 1. Plugging in the value of cos(θ), we get sin²(θ) + (1/3)² = 1. Solving for sin(θ), we find sin(θ) = -√8/3, which is negative since sin(θ) < 0.

Using the values of sin(θ) and cos(θ), we can find the values of the remaining trigonometric functions. tan(θ) = sin(θ)/cos(θ), csc(θ) = 1/sin(θ), sec(θ) = 1/cos(θ), and cot(θ) = 1/tan(θ). Substituting in the values, we get:

sin(θ) = -√8/3
cos(θ) = 1/3
tan(θ) = -√8
csc(θ) = -√3/√8
sec(θ) = 3
cot(θ) = -1/√8