Trigonometry Examples

Find the Other Trig Values in Quadrant II sin(theta)=(2 square root of 5)/5
Step 1
Use the definition of sine 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
Simplify inside the radical.
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Step 4.1
Negate .
Adjacent
Step 4.2
Raise to the power of .
Adjacent
Step 4.3
Apply the product rule to .
Adjacent
Step 4.4
Raise to the power of .
Adjacent
Step 4.5
Rewrite as .
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Step 4.5.1
Use to rewrite as .
Adjacent
Step 4.5.2
Apply the power rule and multiply exponents, .
Adjacent
Step 4.5.3
Combine and .
Adjacent
Step 4.5.4
Cancel the common factor of .
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Step 4.5.4.1
Cancel the common factor.
Adjacent
Step 4.5.4.2
Rewrite the expression.
Adjacent
Adjacent
Step 4.5.5
Evaluate the exponent.
Adjacent
Adjacent
Step 4.6
Multiply .
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Step 4.6.1
Multiply by .
Adjacent
Step 4.6.2
Multiply by .
Adjacent
Adjacent
Step 4.7
Subtract from .
Adjacent
Adjacent
Step 5
Find the value of cosine.
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Step 5.1
Use the definition of cosine 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 tangent.
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Step 6.1
Use the definition of tangent 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
Cancel the common factor of .
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Step 6.3.1.1
Cancel the common factor.
Step 6.3.1.2
Rewrite the expression.
Step 6.3.1.3
Move the negative one from the denominator of .
Step 6.3.2
Multiply by .
Step 7
Find the value of cotangent.
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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 .
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Step 7.3.1
Cancel the common factor of .
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Step 7.3.1.1
Cancel the common factor.
Step 7.3.1.2
Rewrite the expression.
Step 7.3.2
Move the negative in front of the fraction.
Step 8
Find the value of secant.
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Step 8.1
Use the definition of secant to find the value of .
Step 8.2
Substitute in the known values.
Step 8.3
Simplify the value of .
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Step 8.3.1
Move the negative in front of the fraction.
Step 8.3.2
Multiply by .
Step 8.3.3
Combine and simplify the denominator.
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Step 8.3.3.1
Multiply by .
Step 8.3.3.2
Raise to the power of .
Step 8.3.3.3
Raise to the power of .
Step 8.3.3.4
Use the power rule to combine exponents.
Step 8.3.3.5
Add and .
Step 8.3.3.6
Rewrite as .
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Step 8.3.3.6.1
Use to rewrite as .
Step 8.3.3.6.2
Apply the power rule and multiply exponents, .
Step 8.3.3.6.3
Combine and .
Step 8.3.3.6.4
Cancel the common factor of .
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Step 8.3.3.6.4.1
Cancel the common factor.
Step 8.3.3.6.4.2
Rewrite the expression.
Step 8.3.3.6.5
Evaluate the exponent.
Step 8.3.4
Cancel the common factor of .
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Step 8.3.4.1
Cancel the common factor.
Step 8.3.4.2
Divide by .
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
Simplify the value of .
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Step 9.3.1
Multiply by .
Step 9.3.2
Combine and simplify the denominator.
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Step 9.3.2.1
Multiply by .
Step 9.3.2.2
Move .
Step 9.3.2.3
Raise to the power of .
Step 9.3.2.4
Raise to the power of .
Step 9.3.2.5
Use the power rule to combine exponents.
Step 9.3.2.6
Add and .
Step 9.3.2.7
Rewrite as .
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Step 9.3.2.7.1
Use to rewrite as .
Step 9.3.2.7.2
Apply the power rule and multiply exponents, .
Step 9.3.2.7.3
Combine and .
Step 9.3.2.7.4
Cancel the common factor of .
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Step 9.3.2.7.4.1
Cancel the common factor.
Step 9.3.2.7.4.2
Rewrite the expression.
Step 9.3.2.7.5
Evaluate the exponent.
Step 9.3.3
Cancel the common factor of .
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Step 9.3.3.1
Cancel the common factor.
Step 9.3.3.2
Rewrite the expression.
Step 10
This is the solution to each trig value.