Suppose tan(θ) is undefined and sin(θ) < 0. Find the exact values of the 6 trigonometric functions of the angle θ .

Question

Suppose tan(θ) is undefined and sin(θ) < 0. Find the exact values of the 6 trigonometric functions of the angle θ.

Answer 1

Hello Carson,

Using the trig identity tan(θ) = sin(θ) / cos(θ) = y / x on the unit circle,

tan(θ) will be undefined when x = 0, since you would be dividing by zero.

 

This places you at an angle of either 90º or 270º.

 

Since we are told sin(θ) < 0 (which is to say, the y coordinate is negative), we must be at 270º.

 

The coordinate there is (0 , -1)

 

Therefore the values of the six trig functions are:

 

sin(θ) = y = -1

cos(θ) = x = 0

tan(θ) = undefined

 

csc(θ) = 1 / sin(θ) = 1 / -1 = -1

sec(θ) = 1 / cos(θ) = 1/0 = undefined

cot(θ) = cos(θ) / sin(θ) = x / y = 0 / -1 = 0

 

Hope the above helps.  Should you have any further question on it, please feel free to message me directly.

 

Kind regards,

Michael E.

Answer 2

If tanθ is undefined, then

 

cosθ = 0

secθ = 1/cosθ is undefined

 

sinθ = 1 or -1, but sinθ < 0, so

 

sinθ = -1

cscθ = 1/sinθ = 1/-1 = -1

cotθ = cosθ/sinθ = 0/-1 = 0