Trigonometry Examples

Find the Trig Value sin(x)=8/9 , cos(2x)
,
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
Raise to the power of .
Adjacent
Step 4.2
Raise to the power of .
Adjacent
Step 4.3
Multiply by .
Adjacent
Step 4.4
Subtract from .
Adjacent
Adjacent
Step 5
Use the definition of cosine to find the value of .
Step 6
Substitute in the known values.
Step 7
Use the double-angle identity to transform to .
Step 8
Use the definition of to find the value of . In this case, .
Step 9
Substitute the values into .
Step 10
Evaluate to find .
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Step 10.1
Simplify each term.
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Step 10.1.1
Apply the product rule to .
Step 10.1.2
Rewrite as .
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Step 10.1.2.1
Use to rewrite as .
Step 10.1.2.2
Apply the power rule and multiply exponents, .
Step 10.1.2.3
Combine and .
Step 10.1.2.4
Cancel the common factor of .
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Step 10.1.2.4.1
Cancel the common factor.
Step 10.1.2.4.2
Rewrite the expression.
Step 10.1.2.5
Evaluate the exponent.
Step 10.1.3
Raise to the power of .
Step 10.1.4
Multiply .
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Step 10.1.4.1
Combine and .
Step 10.1.4.2
Multiply by .
Step 10.2
To write as a fraction with a common denominator, multiply by .
Step 10.3
Combine and .
Step 10.4
Combine the numerators over the common denominator.
Step 10.5
Simplify the numerator.
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Step 10.5.1
Multiply by .
Step 10.5.2
Subtract from .
Step 10.6
Move the negative in front of the fraction.