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Trigonometry Examples
Step 1
Step 1.1
Factor out of .
Step 1.2
Apply the sine double-angle identity.
Step 1.3
Multiply by .
Step 1.4
Use the double-angle identity to transform to .
Step 1.5
Apply the distributive property.
Step 1.6
Multiply by .
Step 1.7
Multiply by by adding the exponents.
Step 1.7.1
Move .
Step 1.7.2
Multiply by .
Step 1.7.2.1
Raise to the power of .
Step 1.7.2.2
Use the power rule to combine exponents.
Step 1.7.3
Add and .
Step 1.8
Multiply by .
Step 1.9
Apply the sine double-angle identity.
Step 1.10
Multiply by .
Step 2
Step 2.1
Factor out of .
Step 2.1.1
Factor out of .
Step 2.1.2
Factor out of .
Step 2.1.3
Factor out of .
Step 2.1.4
Factor out of .
Step 2.1.5
Factor out of .
Step 2.2
Reorder terms.
Step 3
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 4
Step 4.1
Set equal to .
Step 4.2
Solve for .
Step 4.2.1
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 4.2.2
Simplify the right side.
Step 4.2.2.1
The exact value of is .
Step 4.2.3
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 4.2.4
Subtract from .
Step 4.2.5
Find the period of .
Step 4.2.5.1
The period of the function can be calculated using .
Step 4.2.5.2
Replace with in the formula for period.
Step 4.2.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 4.2.5.4
Divide by .
Step 4.2.6
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 5
Step 5.1
Set equal to .
Step 5.2
Solve for .
Step 5.2.1
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 5.2.2
Simplify the right side.
Step 5.2.2.1
The exact value of is .
Step 5.2.3
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 5.2.4
Simplify .
Step 5.2.4.1
To write as a fraction with a common denominator, multiply by .
Step 5.2.4.2
Combine fractions.
Step 5.2.4.2.1
Combine and .
Step 5.2.4.2.2
Combine the numerators over the common denominator.
Step 5.2.4.3
Simplify the numerator.
Step 5.2.4.3.1
Multiply by .
Step 5.2.4.3.2
Subtract from .
Step 5.2.5
Find the period of .
Step 5.2.5.1
The period of the function can be calculated using .
Step 5.2.5.2
Replace with in the formula for period.
Step 5.2.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 5.2.5.4
Divide by .
Step 5.2.6
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 6
Step 6.1
Set equal to .
Step 6.2
Solve for .
Step 6.2.1
Move all terms not containing to the right side of the equation.
Step 6.2.1.1
Subtract from both sides of the equation.
Step 6.2.1.2
Add to both sides of the equation.
Step 6.2.2
Divide each term in by and simplify.
Step 6.2.2.1
Divide each term in by .
Step 6.2.2.2
Simplify the left side.
Step 6.2.2.2.1
Cancel the common factor of .
Step 6.2.2.2.1.1
Cancel the common factor.
Step 6.2.2.2.1.2
Divide by .
Step 6.2.2.3
Simplify the right side.
Step 6.2.2.3.1
Simplify each term.
Step 6.2.2.3.1.1
Cancel the common factor of and .
Step 6.2.2.3.1.1.1
Factor out of .
Step 6.2.2.3.1.1.2
Cancel the common factors.
Step 6.2.2.3.1.1.2.1
Factor out of .
Step 6.2.2.3.1.1.2.2
Cancel the common factor.
Step 6.2.2.3.1.1.2.3
Rewrite the expression.
Step 6.2.2.3.1.2
Move the negative in front of the fraction.
Step 6.2.3
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 6.2.4
Simplify .
Step 6.2.4.1
To write as a fraction with a common denominator, multiply by .
Step 6.2.4.2
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Step 6.2.4.2.1
Multiply by .
Step 6.2.4.2.2
Multiply by .
Step 6.2.4.3
Combine the numerators over the common denominator.
Step 6.2.4.4
Rewrite as .
Step 6.2.4.5
Simplify the denominator.
Step 6.2.4.5.1
Rewrite as .
Step 6.2.4.5.2
Pull terms out from under the radical, assuming positive real numbers.
Step 6.2.5
The complete solution is the result of both the positive and negative portions of the solution.
Step 6.2.5.1
First, use the positive value of the to find the first solution.
Step 6.2.5.2
Next, use the negative value of the to find the second solution.
Step 6.2.5.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 6.2.6
Set up each of the solutions to solve for .
Step 6.2.7
Solve for in .
Step 6.2.7.1
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 6.2.7.2
Simplify the right side.
Step 6.2.7.2.1
Evaluate .
Step 6.2.7.3
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 6.2.7.4
Simplify .
Step 6.2.7.4.1
To write as a fraction with a common denominator, multiply by .
Step 6.2.7.4.2
Combine fractions.
Step 6.2.7.4.2.1
Combine and .
Step 6.2.7.4.2.2
Combine the numerators over the common denominator.
Step 6.2.7.4.3
Simplify the numerator.
Step 6.2.7.4.3.1
Move to the left of .
Step 6.2.7.4.3.2
Subtract from .
Step 6.2.7.5
Find the period of .
Step 6.2.7.5.1
The period of the function can be calculated using .
Step 6.2.7.5.2
Replace with in the formula for period.
Step 6.2.7.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 6.2.7.5.4
Divide by .
Step 6.2.7.6
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Step 6.2.8
Solve for in .
Step 6.2.8.1
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 6.2.8.2
Simplify the right side.
Step 6.2.8.2.1
Evaluate .
Step 6.2.8.3
The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from , to find a reference angle. Next, add this reference angle to to find the solution in the third quadrant.
Step 6.2.8.4
Simplify the expression to find the second solution.
Step 6.2.8.4.1
Subtract from .
Step 6.2.8.4.2
The resulting angle of is positive, less than , and coterminal with .
Step 6.2.8.5
Find the period of .
Step 6.2.8.5.1
The period of the function can be calculated using .
Step 6.2.8.5.2
Replace with in the formula for period.
Step 6.2.8.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 6.2.8.5.4
Divide by .
Step 6.2.8.6
Add to every negative angle to get positive angles.
Step 6.2.8.6.1
Add to to find the positive angle.
Step 6.2.8.6.2
To write as a fraction with a common denominator, multiply by .
Step 6.2.8.6.3
Combine fractions.
Step 6.2.8.6.3.1
Combine and .
Step 6.2.8.6.3.2
Combine the numerators over the common denominator.
Step 6.2.8.6.4
Simplify the numerator.
Step 6.2.8.6.4.1
Multiply by .
Step 6.2.8.6.4.2
Subtract from .
Step 6.2.8.6.5
List the new angles.
Step 6.2.8.7
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Step 6.2.9
List all of the solutions.
, for any integer
Step 6.2.10
Consolidate the solutions.
Step 6.2.10.1
Consolidate and to .
, for any integer
Step 6.2.10.2
Consolidate and to .
, for any integer
, for any integer
, for any integer
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Step 7
The final solution is all the values that make true.
, for any integer
Step 8
Step 8.1
Consolidate and to .
, for any integer
Step 8.2
Consolidate and to .
, for any integer
Step 8.3
Consolidate and to .
, for any integer
, for any integer