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Find the values of the trigonometric functions of from the information given. cos() = − 10/13 , tan() < 0

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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)

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