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Trigonometry Examples
Step 1
Use the definition of cosecant 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 adjacent side of the unit circle triangle. Since the hypotenuse and opposite 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
Rewrite as .
Step 4.1.1
Use to rewrite as .
Adjacent
Step 4.1.2
Apply the power rule and multiply exponents, .
Adjacent
Step 4.1.3
Combine and .
Adjacent
Step 4.1.4
Cancel the common factor of .
Step 4.1.4.1
Cancel the common factor.
Adjacent
Step 4.1.4.2
Rewrite the expression.
Adjacent
Adjacent
Step 4.1.5
Evaluate the exponent.
Adjacent
Adjacent
Step 4.2
Multiply by by adding the exponents.
Step 4.2.1
Multiply by .
Step 4.2.1.1
Raise to the power of .
Adjacent
Step 4.2.1.2
Use the power rule to combine exponents.
Adjacent
Adjacent
Step 4.2.2
Add and .
Adjacent
Adjacent
Step 4.3
Raise to the power of .
Adjacent
Step 4.4
Subtract from .
Adjacent
Adjacent
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
Simplify the value of .
Step 5.3.1
Move the negative in front of the fraction.
Step 5.3.2
Multiply by .
Step 5.3.3
Combine and simplify the denominator.
Step 5.3.3.1
Multiply by .
Step 5.3.3.2
Raise to the power of .
Step 5.3.3.3
Raise to the power of .
Step 5.3.3.4
Use the power rule to combine exponents.
Step 5.3.3.5
Add and .
Step 5.3.3.6
Rewrite as .
Step 5.3.3.6.1
Use to rewrite as .
Step 5.3.3.6.2
Apply the power rule and multiply exponents, .
Step 5.3.3.6.3
Combine and .
Step 5.3.3.6.4
Cancel the common factor of .
Step 5.3.3.6.4.1
Cancel the common factor.
Step 5.3.3.6.4.2
Rewrite the expression.
Step 5.3.3.6.5
Evaluate the exponent.
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
Simplify the value of .
Step 6.3.1
Multiply by .
Step 6.3.2
Combine and simplify the denominator.
Step 6.3.2.1
Multiply by .
Step 6.3.2.2
Raise to the power of .
Step 6.3.2.3
Raise to the power of .
Step 6.3.2.4
Use the power rule to combine exponents.
Step 6.3.2.5
Add and .
Step 6.3.2.6
Rewrite as .
Step 6.3.2.6.1
Use to rewrite as .
Step 6.3.2.6.2
Apply the power rule and multiply exponents, .
Step 6.3.2.6.3
Combine and .
Step 6.3.2.6.4
Cancel the common factor of .
Step 6.3.2.6.4.1
Cancel the common factor.
Step 6.3.2.6.4.2
Rewrite the expression.
Step 6.3.2.6.5
Evaluate the exponent.
Step 6.3.3
Simplify the numerator.
Step 6.3.3.1
Combine using the product rule for radicals.
Step 6.3.3.2
Multiply by .
Step 7
Step 7.1
Use the definition of tangent to find the value of .
Step 7.2
Substitute in the known values.
Step 7.3
Simplify the value of .
Step 7.3.1
Move the negative in front of the fraction.
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 cotangent to find the value of .
Step 8.2
Substitute in the known values.
Step 8.3
Simplify the value of .
Step 8.3.1
Move the negative one from the denominator of .
Step 8.3.2
Rewrite as .
Step 9
Step 9.1
Use the definition of secant to find the value of .
Step 9.2
Substitute in the known values.
Step 9.3
Simplify the value of .
Step 9.3.1
Multiply by .
Step 9.3.2
Combine and simplify the denominator.
Step 9.3.2.1
Multiply by .
Step 9.3.2.2
Raise to the power of .
Step 9.3.2.3
Raise to the power of .
Step 9.3.2.4
Use the power rule to combine exponents.
Step 9.3.2.5
Add and .
Step 9.3.2.6
Rewrite as .
Step 9.3.2.6.1
Use to rewrite as .
Step 9.3.2.6.2
Apply the power rule and multiply exponents, .
Step 9.3.2.6.3
Combine and .
Step 9.3.2.6.4
Cancel the common factor of .
Step 9.3.2.6.4.1
Cancel the common factor.
Step 9.3.2.6.4.2
Rewrite the expression.
Step 9.3.2.6.5
Evaluate the exponent.
Step 9.3.3
Simplify the numerator.
Step 9.3.3.1
Combine using the product rule for radicals.
Step 9.3.3.2
Multiply by .
Step 10
This is the solution to each trig value.