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.

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.

Replace the known values in the equation.

Raise <math><mstyle displaystyle="true"><mn>24</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Hypotenuse <math><mstyle displaystyle="true"><mo>=</mo><msqrt><mn>576</mn><mo>+</mo><msup><mrow><mo>(</mo><mn>7</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></msqrt></mstyle></math>

Raise <math><mstyle displaystyle="true"><mn>7</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Hypotenuse <math><mstyle displaystyle="true"><mo>=</mo><msqrt><mn>576</mn><mo>+</mo><mn>49</mn></msqrt></mstyle></math>

Add <math><mstyle displaystyle="true"><mn>576</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>49</mn></mstyle></math> .

Hypotenuse <math><mstyle displaystyle="true"><mo>=</mo><msqrt><mn>625</mn></msqrt></mstyle></math>

Rewrite <math><mstyle displaystyle="true"><mn>625</mn></mstyle></math> as <math><mstyle displaystyle="true"><msup><mrow><mn>25</mn></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> .

Hypotenuse <math><mstyle displaystyle="true"><mo>=</mo><msqrt><msup><mrow><mn>25</mn></mrow><mrow><mn>2</mn></mrow></msup></msqrt></mstyle></math>

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

Hypotenuse <math><mstyle displaystyle="true"><mo>=</mo><mn>25</mn></mstyle></math>

Hypotenuse <math><mstyle displaystyle="true"><mo>=</mo><mn>25</mn></mstyle></math>

Use the definition of sine to find the value of <math><mstyle displaystyle="true"><mi>sin</mi><mrow><mo>(</mo><mi>θ</mi><mo>)</mo></mrow></mstyle></math> .

Substitute in the known values.

Use the definition of cosine to find the value of <math><mstyle displaystyle="true"><mi>cos</mi><mrow><mo>(</mo><mi>θ</mi><mo>)</mo></mrow></mstyle></math> .

Substitute in the known values.

Use the definition of cotangent to find the value of <math><mstyle displaystyle="true"><mi>cot</mi><mrow><mo>(</mo><mi>θ</mi><mo>)</mo></mrow></mstyle></math> .

Substitute in the known values.

Use the definition of secant to find the value of <math><mstyle displaystyle="true"><mi>sec</mi><mrow><mo>(</mo><mi>θ</mi><mo>)</mo></mrow></mstyle></math> .

Substitute in the known values.

Use the definition of cosecant to find the value of <math><mstyle displaystyle="true"><mi>csc</mi><mrow><mo>(</mo><mi>θ</mi><mo>)</mo></mrow></mstyle></math> .

Substitute in the known values.

This is the solution to each trig value.

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