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Given cosθ= 13/12 and tanθ<0, find the value of the remaining five trigonometric functions.

User Mala
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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

User Willy David Jr
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