Trigonometry Examples

Solve for ? sin(x)+cos(x)=- square root of 2
Step 1
Square both sides of the equation.
Step 2
Simplify .
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Step 2.1
Rewrite as .
Step 2.2
Expand using the FOIL Method.
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Step 2.2.1
Apply the distributive property.
Step 2.2.2
Apply the distributive property.
Step 2.2.3
Apply the distributive property.
Step 2.3
Simplify and combine like terms.
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Step 2.3.1
Simplify each term.
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Step 2.3.1.1
Multiply .
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Step 2.3.1.1.1
Raise to the power of .
Step 2.3.1.1.2
Raise to the power of .
Step 2.3.1.1.3
Use the power rule to combine exponents.
Step 2.3.1.1.4
Add and .
Step 2.3.1.2
Multiply .
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Step 2.3.1.2.1
Raise to the power of .
Step 2.3.1.2.2
Raise to the power of .
Step 2.3.1.2.3
Use the power rule to combine exponents.
Step 2.3.1.2.4
Add and .
Step 2.3.2
Reorder the factors of .
Step 2.3.3
Add and .
Step 2.4
Move .
Step 2.5
Apply pythagorean identity.
Step 2.6
Simplify each term.
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Step 2.6.1
Reorder and .
Step 2.6.2
Reorder and .
Step 2.6.3
Apply the sine double-angle identity.
Step 3
Simplify .
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Step 3.1
Simplify the expression.
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Step 3.1.1
Apply the product rule to .
Step 3.1.2
Raise to the power of .
Step 3.1.3
Multiply by .
Step 3.2
Rewrite as .
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Step 3.2.1
Use to rewrite as .
Step 3.2.2
Apply the power rule and multiply exponents, .
Step 3.2.3
Combine and .
Step 3.2.4
Cancel the common factor of .
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Step 3.2.4.1
Cancel the common factor.
Step 3.2.4.2
Rewrite the expression.
Step 3.2.5
Evaluate the exponent.
Step 4
Move all terms not containing to the right side of the equation.
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Step 4.1
Subtract from both sides of the equation.
Step 4.2
Subtract from .
Step 5
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 6
Simplify the right side.
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Step 6.1
The exact value of is .
Step 7
Divide each term in by and simplify.
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Step 7.1
Divide each term in by .
Step 7.2
Simplify the left side.
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Step 7.2.1
Cancel the common factor of .
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Step 7.2.1.1
Cancel the common factor.
Step 7.2.1.2
Divide by .
Step 7.3
Simplify the right side.
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Step 7.3.1
Multiply the numerator by the reciprocal of the denominator.
Step 7.3.2
Multiply .
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Step 7.3.2.1
Multiply by .
Step 7.3.2.2
Multiply by .
Step 8
The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from to find the solution in the second quadrant.
Step 9
Solve for .
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Step 9.1
Simplify.
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Step 9.1.1
To write as a fraction with a common denominator, multiply by .
Step 9.1.2
Combine and .
Step 9.1.3
Combine the numerators over the common denominator.
Step 9.1.4
Subtract from .
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Step 9.1.4.1
Reorder and .
Step 9.1.4.2
Subtract from .
Step 9.2
Divide each term in by and simplify.
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Step 9.2.1
Divide each term in by .
Step 9.2.2
Simplify the left side.
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Step 9.2.2.1
Cancel the common factor of .
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Step 9.2.2.1.1
Cancel the common factor.
Step 9.2.2.1.2
Divide by .
Step 9.2.3
Simplify the right side.
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Step 9.2.3.1
Multiply the numerator by the reciprocal of the denominator.
Step 9.2.3.2
Multiply .
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Step 9.2.3.2.1
Multiply by .
Step 9.2.3.2.2
Multiply by .
Step 10
Find the period of .
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Step 10.1
The period of the function can be calculated using .
Step 10.2
Replace with in the formula for period.
Step 10.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 10.4
Cancel the common factor of .
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Step 10.4.1
Cancel the common factor.
Step 10.4.2
Divide by .
Step 11
The period of the function is so values will repeat every radians in both directions.
, for any integer
Step 12
Verify each of the solutions by substituting them into and solving.
, for any integer