Find the Trig Value tan(theta)=24/7 , 0<=theta<=pi/2

Find the Trig Value tan(theta)=24/7 , 0<=theta<=pi/2

$\tan (θ) = \frac{24}{7}$ , $0 \leq θ \leq \frac{π}{2}$

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 $= 25$

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.

$\sin (θ) = \frac{24}{25}$

$\sin (θ) = \frac{24}{25}$

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.

$\cos (θ) = \frac{7}{25}$

$\cos (θ) = \frac{7}{25}$

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.

$\cot (θ) = \frac{7}{24}$

$\cot (θ) = \frac{7}{24}$

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.

$\sec (θ) = \frac{25}{7}$

$\sec (θ) = \frac{25}{7}$

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.

$\csc (θ) = \frac{25}{24}$

$\csc (θ) = \frac{25}{24}$

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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