Trigonometry Examples

Solve for x square root of 1-sin(x)*sin(x)=cos(x)
Step 1
To remove the radical on the left side of the equation, square both sides of the equation.
Step 2
Simplify each side of the equation.
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Step 2.1
Use to rewrite as .
Step 2.2
Simplify the left side.
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Step 2.2.1
Simplify .
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Step 2.2.1.1
Simplify terms.
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Step 2.2.1.1.1
Multiply the exponents in .
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Step 2.2.1.1.1.1
Apply the power rule and multiply exponents, .
Step 2.2.1.1.1.2
Cancel the common factor of .
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Step 2.2.1.1.1.2.1
Cancel the common factor.
Step 2.2.1.1.1.2.2
Rewrite the expression.
Step 2.2.1.1.2
Multiply .
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Step 2.2.1.1.2.1
Raise to the power of .
Step 2.2.1.1.2.2
Raise to the power of .
Step 2.2.1.1.2.3
Use the power rule to combine exponents.
Step 2.2.1.1.2.4
Add and .
Step 2.2.1.2
Apply pythagorean identity.
Step 2.2.1.3
Multiply the exponents in .
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Step 2.2.1.3.1
Apply the power rule and multiply exponents, .
Step 2.2.1.3.2
Multiply by .
Step 3
Solve for .
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Step 3.1
Since the exponents are equal, the bases of the exponents on both sides of the equation must be equal.
Step 3.2
Solve for .
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Step 3.2.1
Rewrite the absolute value equation as four equations without absolute value bars.
Step 3.2.2
After simplifying, there are only two unique equations to be solved.
Step 3.2.3
Solve for .
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Step 3.2.3.1
For the two functions to be equal, the arguments of each must be equal.
Step 3.2.3.2
Move all terms containing to the left side of the equation.
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Step 3.2.3.2.1
Subtract from both sides of the equation.
Step 3.2.3.2.2
Subtract from .
Step 3.2.3.3
Since , the equation will always be true.
All real numbers
All real numbers
Step 3.2.4
Solve for .
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Step 3.2.4.1
Move all terms containing to the left side of the equation.
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Step 3.2.4.1.1
Add to both sides of the equation.
Step 3.2.4.1.2
Add and .
Step 3.2.4.2
Divide each term in by and simplify.
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Step 3.2.4.2.1
Divide each term in by .
Step 3.2.4.2.2
Simplify the left side.
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Step 3.2.4.2.2.1
Cancel the common factor of .
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Step 3.2.4.2.2.1.1
Cancel the common factor.
Step 3.2.4.2.2.1.2
Divide by .
Step 3.2.4.2.3
Simplify the right side.
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Step 3.2.4.2.3.1
Divide by .
Step 3.2.4.3
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 3.2.4.4
Simplify the right side.
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Step 3.2.4.4.1
The exact value of is .
Step 3.2.4.5
The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from to find the solution in the fourth quadrant.
Step 3.2.4.6
Simplify .
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Step 3.2.4.6.1
To write as a fraction with a common denominator, multiply by .
Step 3.2.4.6.2
Combine fractions.
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Step 3.2.4.6.2.1
Combine and .
Step 3.2.4.6.2.2
Combine the numerators over the common denominator.
Step 3.2.4.6.3
Simplify the numerator.
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Step 3.2.4.6.3.1
Multiply by .
Step 3.2.4.6.3.2
Subtract from .
Step 3.2.4.7
Find the period of .
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Step 3.2.4.7.1
The period of the function can be calculated using .
Step 3.2.4.7.2
Replace with in the formula for period.
Step 3.2.4.7.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 3.2.4.7.4
Divide by .
Step 3.2.4.8
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
, for any integer
, for any integer
Step 4
Consolidate the answers.
, for any integer