# Trigonometry Examples

Step 1

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.

Step 2

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.

Step 3

Replace the known values in the equation.

Step 4

Step 4.1

Apply the product rule to .

Hypotenuse

Step 4.2

Raise to the power of .

Hypotenuse

Step 4.3

Multiply by .

Hypotenuse

Step 4.4

Rewrite as .

Step 4.4.1

Use to rewrite as .

Hypotenuse

Step 4.4.2

Apply the power rule and multiply exponents, .

Hypotenuse

Step 4.4.3

Combine and .

Hypotenuse

Step 4.4.4

Cancel the common factor of .

Step 4.4.4.1

Cancel the common factor.

Hypotenuse

Step 4.4.4.2

Rewrite the expression.

Hypotenuse

Hypotenuse

Step 4.4.5

Evaluate the exponent.

Hypotenuse

Hypotenuse

Step 4.5

Raise to the power of .

Hypotenuse

Step 4.6

Add and .

Hypotenuse

Step 4.7

Rewrite as .

Hypotenuse

Step 4.8

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

Hypotenuse

Hypotenuse

Step 5

Step 5.1

Use the definition of sine to find the value of .

Step 5.2

Substitute in the known values.

Step 5.3

Move the negative in front of the fraction.

Step 6

Step 6.1

Use the definition of cosine to find the value of .

Step 6.2

Substitute in the known values.

Step 6.3

Move the negative in front of the fraction.

Step 7

Step 7.1

Use the definition of cotangent to find the value of .

Step 7.2

Substitute in the known values.

Step 7.3

Simplify the value of .

Step 7.3.1

Dividing two negative values results in a positive value.

Step 7.3.2

Multiply by .

Step 7.3.3

Combine and simplify the denominator.

Step 7.3.3.1

Multiply by .

Step 7.3.3.2

Raise to the power of .

Step 7.3.3.3

Raise to the power of .

Step 7.3.3.4

Use the power rule to combine exponents.

Step 7.3.3.5

Add and .

Step 7.3.3.6

Rewrite as .

Step 7.3.3.6.1

Use to rewrite as .

Step 7.3.3.6.2

Apply the power rule and multiply exponents, .

Step 7.3.3.6.3

Combine and .

Step 7.3.3.6.4

Cancel the common factor of .

Step 7.3.3.6.4.1

Cancel the common factor.

Step 7.3.3.6.4.2

Rewrite the expression.

Step 7.3.3.6.5

Evaluate the exponent.

Step 8

Step 8.1

Use the definition of secant to find the value of .

Step 8.2

Substitute in the known values.

Step 8.3

Divide by .

Step 9

Step 9.1

Use the definition of cosecant to find the value of .

Step 9.2

Substitute in the known values.

Step 9.3

Simplify the value of .

Step 9.3.1

Move the negative in front of the fraction.

Step 9.3.2

Multiply by .

Step 9.3.3

Combine and simplify the denominator.

Step 9.3.3.1

Multiply by .

Step 9.3.3.2

Raise to the power of .

Step 9.3.3.3

Raise to the power of .

Step 9.3.3.4

Use the power rule to combine exponents.

Step 9.3.3.5

Add and .

Step 9.3.3.6

Rewrite as .

Step 9.3.3.6.1

Use to rewrite as .

Step 9.3.3.6.2

Apply the power rule and multiply exponents, .

Step 9.3.3.6.3

Combine and .

Step 9.3.3.6.4

Cancel the common factor of .

Step 9.3.3.6.4.1

Cancel the common factor.

Step 9.3.3.6.4.2

Rewrite the expression.

Step 9.3.3.6.5

Evaluate the exponent.

Step 10

This is the solution to each trig value.