Find Trig Functions Using Identities tan(theta)=24/7 , sin(theta)<0

$\tan (θ) = \frac{24}{7}$ , $\sin (θ) < 0$

The sine function is negative in the third and fourth quadrants. The tangent function is positive in the first and third quadrants. The set of solutions for $θ$ are limited to the third quadrant since that is the only quadrant found in both sets.

Solution is in the third quadrant.

Use the definition of tangent to find the known sides of the unit circle right triangle. The quadrant determines the sign on each of the values.

$\tan (θ) = \frac{opposite}{adjacent}$

Find the hypotenuse of the unit circle triangle. Since the opposite and adjacent sides are known, use the Pythagorean theorem to find the remaining side.

$Hypotenuse = \sqrt{{opposite}^{2} + {adjacent}^{2}}$

Replace the known values in the equation.

$Hypotenuse = \sqrt{{(- 24)}^{2} + {(- 7)}^{2}}$

Simplify inside the radical.

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Raise $- 24$ to the power of $2$ .

Hypotenuse $= \sqrt{576 + {(- 7)}^{2}}$

Raise $- 7$ to the power of $2$ .

Hypotenuse $= \sqrt{576 + 49}$

Add $576$ and $49$ .

Hypotenuse $= \sqrt{625}$

Rewrite $625$ as $25^{2}$ .

Hypotenuse $= \sqrt{25^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

Hypotenuse $= 25$

Find the value of sine.

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Use the definition of sine to find the value of $\sin (θ)$ .

$\sin (θ) = \frac{opp}{hyp}$

Substitute in the known values.

$\sin (θ) = \frac{- 24}{25}$

Move the negative in front of the fraction.

$\sin (θ) = - \frac{24}{25}$

Find the value of cosine.

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Use the definition of cosine to find the value of $\cos (θ)$ .

$\cos (θ) = \frac{adj}{hyp}$

Substitute in the known values.

$\cos (θ) = \frac{- 7}{25}$

Move the negative in front of the fraction.

$\cos (θ) = - \frac{7}{25}$

Find the value of cotangent.

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Use the definition of cotangent to find the value of $\cot (θ)$ .

$\cot (θ) = \frac{adj}{opp}$

Substitute in the known values.

$\cot (θ) = \frac{- 7}{- 24}$

Dividing two negative values results in a positive value.

$\cot (θ) = \frac{7}{24}$

Find the value of secant.

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Use the definition of secant to find the value of $\sec (θ)$ .

$\sec (θ) = \frac{hyp}{adj}$

Substitute in the known values.

$\sec (θ) = \frac{25}{- 7}$

Move the negative in front of the fraction.

$\sec (θ) = - \frac{25}{7}$

Find the value of cosecant.

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Use the definition of cosecant to find the value of $\csc (θ)$ .

$\csc (θ) = \frac{hyp}{opp}$

Substitute in the known values.

$\csc (θ) = \frac{25}{- 24}$

Move the negative in front of the fraction.

$\csc (θ) = - \frac{25}{24}$

This is the solution to each trig value.

$\sin (θ) = - \frac{24}{25}$

$\cos (θ) = - \frac{7}{25}$

$\tan (θ) = \frac{24}{7}$

$\cot (θ) = \frac{7}{24}$

$\sec (θ) = - \frac{25}{7}$

$\csc (θ) = - \frac{25}{24}$

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