Trigonometry Examples

Solve for x sin(2x)+ square root of 2cos(x)=0
Step 1
Subtract from both sides of the equation.
Step 2
To remove the radical on the left side of the equation, square both sides of the equation.
Step 3
Simplify each side of the equation.
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Step 3.1
Use to rewrite as .
Step 3.2
Simplify the left side.
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Step 3.2.1
Simplify .
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Step 3.2.1.1
Multiply the exponents in .
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Step 3.2.1.1.1
Apply the power rule and multiply exponents, .
Step 3.2.1.1.2
Cancel the common factor of .
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Step 3.2.1.1.2.1
Cancel the common factor.
Step 3.2.1.1.2.2
Rewrite the expression.
Step 3.2.1.2
Simplify.
Step 3.3
Simplify the right side.
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Step 3.3.1
Simplify .
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Step 3.3.1.1
Apply the product rule to .
Step 3.3.1.2
Raise to the power of .
Step 3.3.1.3
Multiply by .
Step 4
Solve for .
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Step 4.1
Subtract from both sides of the equation.
Step 4.2
Replace with .
Step 4.3
Simplify the left side of the equation.
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Step 4.3.1
Apply pythagorean identity.
Step 4.3.2
Simplify each term.
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Step 4.3.2.1
Apply the sine double-angle identity.
Step 4.3.2.2
Use the power rule to distribute the exponent.
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Step 4.3.2.2.1
Apply the product rule to .
Step 4.3.2.2.2
Apply the product rule to .
Step 4.3.2.3
Raise to the power of .
Step 4.3.2.4
Multiply by .
Step 4.4
Factor out of .
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Step 4.4.1
Factor out of .
Step 4.4.2
Factor out of .
Step 4.4.3
Factor out of .
Step 4.5
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 4.6
Set equal to and solve for .
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Step 4.6.1
Set equal to .
Step 4.6.2
Solve for .
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Step 4.6.2.1
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 4.6.2.2
Simplify the right side.
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Step 4.6.2.2.1
The exact value of is .
Step 4.6.2.3
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 4.6.2.4
Simplify .
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Step 4.6.2.4.1
To write as a fraction with a common denominator, multiply by .
Step 4.6.2.4.2
Combine fractions.
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Step 4.6.2.4.2.1
Combine and .
Step 4.6.2.4.2.2
Combine the numerators over the common denominator.
Step 4.6.2.4.3
Simplify the numerator.
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Step 4.6.2.4.3.1
Multiply by .
Step 4.6.2.4.3.2
Subtract from .
Step 4.6.2.5
Find the period of .
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Step 4.6.2.5.1
The period of the function can be calculated using .
Step 4.6.2.5.2
Replace with in the formula for period.
Step 4.6.2.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 4.6.2.5.4
Divide by .
Step 4.6.2.6
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
, for any integer
Step 4.7
Set equal to and solve for .
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Step 4.7.1
Set equal to .
Step 4.7.2
Solve for .
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Step 4.7.2.1
Replace the with based on the identity.
Step 4.7.2.2
Multiply by .
Step 4.7.2.3
Apply the distributive property.
Step 4.7.2.4
Multiply by .
Step 4.7.2.5
Multiply by by adding the exponents.
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Step 4.7.2.5.1
Move .
Step 4.7.2.5.2
Multiply by .
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Step 4.7.2.5.2.1
Raise to the power of .
Step 4.7.2.5.2.2
Use the power rule to combine exponents.
Step 4.7.2.5.3
Add and .
Step 4.7.2.6
Reorder the polynomial.
Step 4.7.2.7
Substitute for .
Step 4.7.2.8
Factor the left side of the equation.
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Step 4.7.2.8.1
Factor out of .
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Step 4.7.2.8.1.1
Factor out of .
Step 4.7.2.8.1.2
Rewrite as .
Step 4.7.2.8.1.3
Factor out of .
Step 4.7.2.8.1.4
Factor out of .
Step 4.7.2.8.2
Rewrite as .
Step 4.7.2.8.3
Factor.
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Step 4.7.2.8.3.1
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 4.7.2.8.3.2
Remove unnecessary parentheses.
Step 4.7.2.9
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 4.7.2.10
Set equal to .
Step 4.7.2.11
Set equal to and solve for .
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Step 4.7.2.11.1
Set equal to .
Step 4.7.2.11.2
Subtract from both sides of the equation.
Step 4.7.2.12
Set equal to and solve for .
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Step 4.7.2.12.1
Set equal to .
Step 4.7.2.12.2
Add to both sides of the equation.
Step 4.7.2.13
The final solution is all the values that make true.
Step 4.7.2.14
Substitute for .
Step 4.7.2.15
Set up each of the solutions to solve for .
Step 4.7.2.16
Solve for in .
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Step 4.7.2.16.1
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 4.7.2.16.2
Simplify the right side.
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Step 4.7.2.16.2.1
The exact value of is .
Step 4.7.2.16.3
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 4.7.2.16.4
Simplify .
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Step 4.7.2.16.4.1
To write as a fraction with a common denominator, multiply by .
Step 4.7.2.16.4.2
Combine fractions.
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Step 4.7.2.16.4.2.1
Combine and .
Step 4.7.2.16.4.2.2
Combine the numerators over the common denominator.
Step 4.7.2.16.4.3
Simplify the numerator.
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Step 4.7.2.16.4.3.1
Multiply by .
Step 4.7.2.16.4.3.2
Subtract from .
Step 4.7.2.16.5
Find the period of .
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Step 4.7.2.16.5.1
The period of the function can be calculated using .
Step 4.7.2.16.5.2
Replace with in the formula for period.
Step 4.7.2.16.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 4.7.2.16.5.4
Divide by .
Step 4.7.2.16.6
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Step 4.7.2.17
Solve for in .
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Step 4.7.2.17.1
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 4.7.2.17.2
Simplify the right side.
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Step 4.7.2.17.2.1
The exact value of is .
Step 4.7.2.17.3
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 4.7.2.17.4
Subtract from .
Step 4.7.2.17.5
Find the period of .
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Step 4.7.2.17.5.1
The period of the function can be calculated using .
Step 4.7.2.17.5.2
Replace with in the formula for period.
Step 4.7.2.17.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 4.7.2.17.5.4
Divide by .
Step 4.7.2.17.6
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Step 4.7.2.18
Solve for in .
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Step 4.7.2.18.1
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 4.7.2.18.2
Simplify the right side.
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Step 4.7.2.18.2.1
The exact value of is .
Step 4.7.2.18.3
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 4.7.2.18.4
Subtract from .
Step 4.7.2.18.5
Find the period of .
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Step 4.7.2.18.5.1
The period of the function can be calculated using .
Step 4.7.2.18.5.2
Replace with in the formula for period.
Step 4.7.2.18.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 4.7.2.18.5.4
Divide by .
Step 4.7.2.18.6
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Step 4.7.2.19
List all of the solutions.
, for any integer
Step 4.7.2.20
Consolidate the answers.
, for any integer
, for any integer
, for any integer
Step 4.8
The final solution is all the values that make true.
, for any integer
, for any integer
Step 5
Consolidate the answers.
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Step 5.1
Consolidate and to .
, for any integer
Step 5.2
Consolidate and to .
, for any integer
, for any integer
Step 6
Verify each of the solutions by substituting them into and solving.
, for any integer