# Trigonometry Examples

Find the Other Trig Values in Quadrant I
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
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
Find the value of sine.
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
Find the value of cosine.
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
Find the value of cotangent.
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
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
Find the value of secant.
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
Find the value of cosecant.
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
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.