Answer:
To find the values of the remaining five trigonometric functions, we can use the given information about the cosine (cosθ) and tangent (tanθ) of θ.
Given:
cosθ = 13/12
tanθ < 0
We can start by finding the value of sinθ using the Pythagorean identity:
sinθ = √(1 - cos²θ)
Given cosθ = 13/12, we can substitute it into the formula:
sinθ = √(1 - (13/12)²)
sinθ = √(1 - 169/144)
sinθ = √(144/144 - 169/144)
sinθ = √((-25)/144)
sinθ = -5/12
Next, we can find the value of secθ using the reciprocal of cosθ:
secθ = 1/cosθ
secθ = 1/(13/12)
secθ = 12/13
Since tanθ < 0 and tanθ = sinθ/cosθ, we know that sinθ and cosθ have opposite signs. Since sinθ is negative, cosθ must be positive.
Now, we can determine the values of the remaining three trigonometric functions:
cscθ = 1/sinθ
cscθ = 1/(-5/12)
cscθ = -12/5
cotθ = 1/tanθ
cotθ = 1/(sinθ/cosθ)
cotθ = cosθ/sinθ
cotθ = (13/12)/(-5/12)
cotθ = -13/5
Since secθ > 0 and cosθ > 0, we can determine the sign of cosecθ and cotθ based on the quadrants where θ lies. In this case, since tanθ < 0, we can infer that θ lies in either the second or fourth quadrant. In these quadrants, cscθ and cotθ are both negative.
Therefore, the values of the remaining trigonometric functions are:
sinθ = -5/12
cscθ = -12/5
secθ = 12/13
cotθ = -13/5