Trigonometry Examples

Find the Other Trig Values in Quadrant III sec(theta)=- square root of 5
Step 1
Use the definition of secant 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 opposite side of the unit circle triangle. Since the adjacent side and hypotenuse are known, use the Pythagorean theorem to find the remaining side.
Step 3
Replace the known values in the equation.
Step 4
Simplify inside the radical.
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Step 4.1
Negate .
Opposite
Step 4.2
Rewrite as .
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Step 4.2.1
Use to rewrite as .
Opposite
Step 4.2.2
Apply the power rule and multiply exponents, .
Opposite
Step 4.2.3
Combine and .
Opposite
Step 4.2.4
Cancel the common factor of .
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Step 4.2.4.1
Cancel the common factor.
Opposite
Step 4.2.4.2
Rewrite the expression.
Opposite
Opposite
Step 4.2.5
Evaluate the exponent.
Opposite
Opposite
Step 4.3
Multiply by by adding the exponents.
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Step 4.3.1
Multiply by .
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Step 4.3.1.1
Raise to the power of .
Opposite
Step 4.3.1.2
Use the power rule to combine exponents.
Opposite
Opposite
Step 4.3.2
Add and .
Opposite
Opposite
Step 4.4
Raise to the power of .
Opposite
Step 4.5
Subtract from .
Opposite
Step 4.6
Rewrite as .
Opposite
Step 4.7
Pull terms out from under the radical, assuming positive real numbers.
Opposite
Step 4.8
Multiply by .
Opposite
Opposite
Step 5
Find the value of sine.
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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 .
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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.
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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 .
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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 .
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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
Find the value of cosine.
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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 .
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Step 6.3.1
Move the negative in front of the fraction.
Step 6.3.2
Multiply by .
Step 6.3.3
Combine and simplify the denominator.
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Step 6.3.3.1
Multiply by .
Step 6.3.3.2
Raise to the power of .
Step 6.3.3.3
Raise to the power of .
Step 6.3.3.4
Use the power rule to combine exponents.
Step 6.3.3.5
Add and .
Step 6.3.3.6
Rewrite as .
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Step 6.3.3.6.1
Use to rewrite as .
Step 6.3.3.6.2
Apply the power rule and multiply exponents, .
Step 6.3.3.6.3
Combine and .
Step 6.3.3.6.4
Cancel the common factor of .
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Step 6.3.3.6.4.1
Cancel the common factor.
Step 6.3.3.6.4.2
Rewrite the expression.
Step 6.3.3.6.5
Evaluate the exponent.
Step 7
Find the value of tangent.
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Step 7.1
Use the definition of tangent to find the value of .
Step 7.2
Substitute in the known values.
Step 7.3
Divide by .
Step 8
Find the value of cotangent.
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Step 8.1
Use the definition of cotangent to find the value of .
Step 8.2
Substitute in the known values.
Step 8.3
Dividing two negative values results in a positive value.
Step 9
Find the value of cosecant.
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Step 9.1
Use the definition of cosecant to find the value of .
Step 9.2
Substitute in the known values.
Step 9.3
Move the negative in front of the fraction.
Step 10
This is the solution to each trig value.