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Trigonometry Examples
Step 1
Step 1.1
Add to both sides of the equation.
Step 1.2
Rewrite in terms of sines and cosines.
Step 1.3
To write as a fraction with a common denominator, multiply by .
Step 1.4
To write as a fraction with a common denominator, multiply by .
Step 1.5
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Step 1.5.1
Multiply by .
Step 1.5.2
Multiply by .
Step 1.5.3
Reorder the factors of .
Step 1.6
Combine the numerators over the common denominator.
Step 1.7
Simplify the numerator.
Step 1.7.1
Apply the distributive property.
Step 1.7.2
Multiply by .
Step 2
Set the numerator equal to zero.
Step 3
Step 3.1
Move all terms not containing to the right side of the equation.
Step 3.1.1
Subtract from both sides of the equation.
Step 3.1.2
Subtract from both sides of the equation.
Step 3.2
To remove the radical on the left side of the equation, square both sides of the equation.
Step 3.3
Simplify each side of the equation.
Step 3.3.1
Use to rewrite as .
Step 3.3.2
Simplify the left side.
Step 3.3.2.1
Simplify .
Step 3.3.2.1.1
Apply the product rule to .
Step 3.3.2.1.2
Multiply the exponents in .
Step 3.3.2.1.2.1
Apply the power rule and multiply exponents, .
Step 3.3.2.1.2.2
Cancel the common factor of .
Step 3.3.2.1.2.2.1
Cancel the common factor.
Step 3.3.2.1.2.2.2
Rewrite the expression.
Step 3.3.2.1.3
Simplify.
Step 3.3.2.1.4
Apply the distributive property.
Step 3.3.2.1.5
Multiply by .
Step 3.3.3
Simplify the right side.
Step 3.3.3.1
Simplify .
Step 3.3.3.1.1
Rewrite as .
Step 3.3.3.1.2
Expand using the FOIL Method.
Step 3.3.3.1.2.1
Apply the distributive property.
Step 3.3.3.1.2.2
Apply the distributive property.
Step 3.3.3.1.2.3
Apply the distributive property.
Step 3.3.3.1.3
Simplify and combine like terms.
Step 3.3.3.1.3.1
Simplify each term.
Step 3.3.3.1.3.1.1
Multiply .
Step 3.3.3.1.3.1.1.1
Multiply by .
Step 3.3.3.1.3.1.1.2
Multiply by .
Step 3.3.3.1.3.1.1.3
Raise to the power of .
Step 3.3.3.1.3.1.1.4
Raise to the power of .
Step 3.3.3.1.3.1.1.5
Use the power rule to combine exponents.
Step 3.3.3.1.3.1.1.6
Add and .
Step 3.3.3.1.3.1.2
Multiply .
Step 3.3.3.1.3.1.2.1
Multiply by .
Step 3.3.3.1.3.1.2.2
Multiply by .
Step 3.3.3.1.3.1.2.3
Raise to the power of .
Step 3.3.3.1.3.1.2.4
Raise to the power of .
Step 3.3.3.1.3.1.2.5
Use the power rule to combine exponents.
Step 3.3.3.1.3.1.2.6
Add and .
Step 3.3.3.1.3.1.3
Multiply .
Step 3.3.3.1.3.1.3.1
Multiply by .
Step 3.3.3.1.3.1.3.2
Multiply by .
Step 3.3.3.1.3.1.3.3
Raise to the power of .
Step 3.3.3.1.3.1.3.4
Raise to the power of .
Step 3.3.3.1.3.1.3.5
Use the power rule to combine exponents.
Step 3.3.3.1.3.1.3.6
Add and .
Step 3.3.3.1.3.1.4
Multiply .
Step 3.3.3.1.3.1.4.1
Multiply by .
Step 3.3.3.1.3.1.4.2
Multiply by .
Step 3.3.3.1.3.1.4.3
Raise to the power of .
Step 3.3.3.1.3.1.4.4
Raise to the power of .
Step 3.3.3.1.3.1.4.5
Use the power rule to combine exponents.
Step 3.3.3.1.3.1.4.6
Add and .
Step 3.3.3.1.3.1.4.7
Raise to the power of .
Step 3.3.3.1.3.1.4.8
Raise to the power of .
Step 3.3.3.1.3.1.4.9
Use the power rule to combine exponents.
Step 3.3.3.1.3.1.4.10
Add and .
Step 3.3.3.1.3.2
Add and .
Step 3.4
Solve for .
Step 3.4.1
Move all the expressions to the left side of the equation.
Step 3.4.1.1
Subtract from both sides of the equation.
Step 3.4.1.2
Subtract from both sides of the equation.
Step 3.4.1.3
Subtract from both sides of the equation.
Step 3.4.2
Simplify .
Step 3.4.2.1
Move .
Step 3.4.2.2
Apply the cosine double-angle identity.
Step 3.4.2.3
Cancel the common factor of .
Step 3.4.2.3.1
Cancel the common factor.
Step 3.4.2.3.2
Rewrite the expression.
Step 3.4.2.4
Multiply by .
Step 3.4.2.5
Factor out of .
Step 3.4.2.6
Factor out of .
Step 3.4.2.7
Rewrite as .
Step 3.4.2.8
Apply pythagorean identity.
Step 3.4.2.9
Reorder the factors of .
Step 3.4.2.10
Subtract from .
Step 3.4.3
Factor out of .
Step 3.4.3.1
Reorder the expression.
Step 3.4.3.1.1
Move .
Step 3.4.3.1.2
Move .
Step 3.4.3.1.3
Reorder and .
Step 3.4.3.2
Factor out of .
Step 3.4.3.3
Factor out of .
Step 3.4.3.4
Factor out of .
Step 3.4.4
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 3.4.5
Set equal to and solve for .
Step 3.4.5.1
Set equal to .
Step 3.4.5.2
Solve for .
Step 3.4.5.2.1
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 3.4.5.2.2
Simplify the right side.
Step 3.4.5.2.2.1
The exact value of is .
Step 3.4.5.2.3
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 3.4.5.2.4
Simplify .
Step 3.4.5.2.4.1
To write as a fraction with a common denominator, multiply by .
Step 3.4.5.2.4.2
Combine fractions.
Step 3.4.5.2.4.2.1
Combine and .
Step 3.4.5.2.4.2.2
Combine the numerators over the common denominator.
Step 3.4.5.2.4.3
Simplify the numerator.
Step 3.4.5.2.4.3.1
Multiply by .
Step 3.4.5.2.4.3.2
Subtract from .
Step 3.4.5.2.5
Find the period of .
Step 3.4.5.2.5.1
The period of the function can be calculated using .
Step 3.4.5.2.5.2
Replace with in the formula for period.
Step 3.4.5.2.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 3.4.5.2.5.4
Divide by .
Step 3.4.5.2.6
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 3.4.6
Set equal to and solve for .
Step 3.4.6.1
Set equal to .
Step 3.4.6.2
Solve for .
Step 3.4.6.2.1
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 3.4.6.2.2
Simplify .
Step 3.4.6.2.2.1
Rewrite as .
Step 3.4.6.2.2.2
Pull terms out from under the radical, assuming positive real numbers.
Step 3.4.6.2.2.3
Plus or minus is .
Step 3.4.6.2.3
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 3.4.6.2.4
Simplify the right side.
Step 3.4.6.2.4.1
The exact value of is .
Step 3.4.6.2.5
Set the numerator equal to zero.
Step 3.4.6.2.6
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 3.4.6.2.7
Solve for .
Step 3.4.6.2.7.1
Multiply both sides of the equation by .
Step 3.4.6.2.7.2
Simplify both sides of the equation.
Step 3.4.6.2.7.2.1
Simplify the left side.
Step 3.4.6.2.7.2.1.1
Cancel the common factor of .
Step 3.4.6.2.7.2.1.1.1
Cancel the common factor.
Step 3.4.6.2.7.2.1.1.2
Rewrite the expression.
Step 3.4.6.2.7.2.2
Simplify the right side.
Step 3.4.6.2.7.2.2.1
Subtract from .
Step 3.4.6.2.8
Find the period of .
Step 3.4.6.2.8.1
The period of the function can be calculated using .
Step 3.4.6.2.8.2
Replace with in the formula for period.
Step 3.4.6.2.8.3
is approximately which is positive so remove the absolute value
Step 3.4.6.2.8.4
Multiply the numerator by the reciprocal of the denominator.
Step 3.4.6.2.8.5
Multiply by .
Step 3.4.6.2.9
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 3.4.7
Set equal to and solve for .
Step 3.4.7.1
Set equal to .
Step 3.4.7.2
Solve for .
Step 3.4.7.2.1
Subtract from both sides of the equation.
Step 3.4.7.2.2
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 3.4.7.2.3
Simplify the right side.
Step 3.4.7.2.3.1
The exact value of is .
Step 3.4.7.2.4
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 3.4.7.2.5
Subtract from .
Step 3.4.7.2.6
Find the period of .
Step 3.4.7.2.6.1
The period of the function can be calculated using .
Step 3.4.7.2.6.2
Replace with in the formula for period.
Step 3.4.7.2.6.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 3.4.7.2.6.4
Divide by .
Step 3.4.7.2.7
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 3.4.8
The final solution is all the values that make true.
, for any integer
, for any integer
, for any integer
Step 4
Step 4.1
Consolidate and to .
, for any integer
Step 4.2
Consolidate and to .
, for any integer
Step 4.3
Consolidate and to .
, for any integer
Step 4.4
Consolidate the answers.
, for any integer
, for any integer
Step 5
Verify each of the solutions by substituting them into and solving.
, for any integer