Answer:
The values of the trigonometric functions are:
sin(θ) = -√69/13
tan(θ) < 0
csc(θ) = 13/√69
sec(θ) = 13/10
cot(θ) = -1/tan(θ)
Explanation:
Given that cos(θ) = -10/13 and tan(θ) < 0, we can determine the values of other trigonometric functions using the information provided.
We know that cos(θ) is negative, which means θ is in either the second or third quadrant of the unit circle. In these quadrants:
In the second quadrant, both sine (sin) and tangent (tan) are positive.
In the third quadrant, only tangent (tan) is positive.
Since tan(θ) < 0, we conclude that θ is in the third quadrant.
Now, let's find the values of other trigonometric functions:
Sine (sin):
In the third quadrant, sine (sin) is also negative. We can use the Pythagorean identity to find sin(θ):
sin(θ) = -√(1 - cos²(θ))
sin(θ) = -√(1 - (-10/13)²) = -√(1 - 100/169) = -√(69/169) = -√69/13
Tangent (tan):
We already know that tan(θ) < 0.
Cosecant (csc):
Since sin(θ) is negative, csc(θ) is the reciprocal of sin(θ) and will also be negative:
csc(θ) = -1/sin(θ) = -1/(-√69/13) = 13/√69
Secant (sec):
Secant (sec) is the reciprocal of cos(θ) and will also be negative:
sec(θ) = -1/cos(θ) = -1/(-10/13) = 13/10
Cotangent (cot):
Cotangent (cot) is the reciprocal of tan(θ) and will also be negative:
cot(θ) = -1/tan(θ) = -1/tan(θ) = -1/tan(θ)