Find all six trigonometric functions of θ if the given point is on the terminal side of θ .(Assume that a is a positive number.)

Question

Find all six trigonometric functions of θ  if the given point is on the terminal side of θ .(Assume that a is a positive number.)  ​

Quizplus · Accepted Answer

To find the six trigonometric functions of an angle, we need to identify the three sides of the right triangle formed by the angle and the origin. In this case, the point is located at (-a,a), so the two legs of the triangle are -a and a, and the hypotenuse is sqrt((-a)^2+a^2) = sqrt(2a^2).

Next, we can use the definitions of the trigonometric functions:

sin θ = opposite/hypotenuse = a/sqrt(2a^2) = sqrt(2)/2
cos θ = adjacent/hypotenuse = -a/sqrt(2a^2) = -sqrt(2)/2
tan θ = opposite/adjacent = -a/a = -1
csc θ = 1/sin θ = sqrt(2)
sec θ = 1/cos θ = -sqrt(2)
cot θ = 1/tan θ = -1

Therefore, the six trigonometric functions of θ are:
sin θ = sqrt(2)/2
cos θ = -sqrt(2)/2
tan θ = -1
csc θ = sqrt(2)
sec θ = -sqrt(2)
cot θ = -1

The only choice that matches these values is B.

Find All Six Trigonometric Functions of θ If the Given