Trigonometry Examples

Solve for x in Radians 4sin(x)^2=5+4cos(x)
Step 1
Move all the expressions to the left side of the equation.
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Step 1.1
Subtract from both sides of the equation.
Step 1.2
Subtract from both sides of the equation.
Step 2
Replace the with based on the identity.
Step 3
Simplify each term.
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Step 3.1
Apply the distributive property.
Step 3.2
Multiply by .
Step 3.3
Multiply by .
Step 4
Subtract from .
Step 5
Reorder the polynomial.
Step 6
Substitute for .
Step 7
Factor the left side of the equation.
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Step 7.1
Factor out of .
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Step 7.1.1
Factor out of .
Step 7.1.2
Factor out of .
Step 7.1.3
Rewrite as .
Step 7.1.4
Factor out of .
Step 7.1.5
Factor out of .
Step 7.2
Factor using the perfect square rule.
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Step 7.2.1
Rewrite as .
Step 7.2.2
Rewrite as .
Step 7.2.3
Check that the middle term is two times the product of the numbers being squared in the first term and third term.
Step 7.2.4
Rewrite the polynomial.
Step 7.2.5
Factor using the perfect square trinomial rule , where and .
Step 8
Divide each term in by and simplify.
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Step 8.1
Divide each term in by .
Step 8.2
Simplify the left side.
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Step 8.2.1
Dividing two negative values results in a positive value.
Step 8.2.2
Divide by .
Step 8.3
Simplify the right side.
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Step 8.3.1
Divide by .
Step 9
Set the equal to .
Step 10
Solve for .
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Step 10.1
Subtract from both sides of the equation.
Step 10.2
Divide each term in by and simplify.
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Step 10.2.1
Divide each term in by .
Step 10.2.2
Simplify the left side.
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Step 10.2.2.1
Cancel the common factor of .
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Step 10.2.2.1.1
Cancel the common factor.
Step 10.2.2.1.2
Divide by .
Step 10.2.3
Simplify the right side.
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Step 10.2.3.1
Move the negative in front of the fraction.
Step 11
Substitute for .
Step 12
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 13
Simplify the right side.
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Step 13.1
The exact value of is .
Step 14
The cosine function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from to find the solution in the third quadrant.
Step 15
Simplify .
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Step 15.1
To write as a fraction with a common denominator, multiply by .
Step 15.2
Combine fractions.
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Step 15.2.1
Combine and .
Step 15.2.2
Combine the numerators over the common denominator.
Step 15.3
Simplify the numerator.
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Step 15.3.1
Multiply by .
Step 15.3.2
Subtract from .
Step 16
Find the period of .
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Step 16.1
The period of the function can be calculated using .
Step 16.2
Replace with in the formula for period.
Step 16.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 16.4
Divide by .
Step 17
The period of the function is so values will repeat every radians in both directions.
, for any integer