# Trigonometry Examples

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.

Rewrite as .

Rewrite as .

Hypotenuse

Apply the power rule and multiply exponents, .

Hypotenuse

Combine and .

Hypotenuse

Cancel the common factor of .

Cancel the common factor.

Hypotenuse

Divide by .

Hypotenuse

Hypotenuse

Evaluate the exponent.

Hypotenuse

Hypotenuse

One to any power is one.

Hypotenuse

Add and .

Hypotenuse

Rewrite as .

Hypotenuse

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

Hypotenuse

Hypotenuse

Use the definition of sine to find the value of .

Substitute in the known values.

Use the definition of cosine to find the value of .

Substitute in the known values.

Use the definition of cotangent to find the value of .

Substitute in the known values.

Simplify the value of .

Multiply by .

Combine and simplify the denominator.

Multiply and .

Raise to the power of .

Raise to the power of .

Use the power rule to combine exponents.

Add and .

Rewrite as .

Rewrite as .

Apply the power rule and multiply exponents, .

Combine and .

Cancel the common factor of .

Cancel the common factor.

Divide by .

Evaluate the exponent.

Use the definition of secant to find the value of .

Substitute in the known values.

Divide by .

Use the definition of cosecant to find the value of .

Substitute in the known values.

Simplify the value of .

Multiply by .

Combine and simplify the denominator.

Multiply and .

Raise to the power of .

Raise to the power of .

Use the power rule to combine exponents.

Add and .

Rewrite as .

Rewrite as .

Apply the power rule and multiply exponents, .

Combine and .

Cancel the common factor of .

Cancel the common factor.

Divide by .

Evaluate the exponent.

This is the solution to each trig value.