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Trigonometry Examples
Step 1
Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.
Step 2
Step 2.1
Cross multiply by setting the product of the numerator of the right side and the denominator of the left side equal to the product of the numerator of the left side and the denominator of the right side.
Step 2.2
Simplify the left side.
Step 2.2.1
Simplify .
Step 2.2.1.1
Rewrite in terms of sines and cosines.
Step 2.2.1.2
Rewrite as .
Step 2.2.1.3
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 2.2.1.4
Combine and .
Step 2.2.1.5
Separate fractions.
Step 2.2.1.6
Convert from to .
Step 2.2.1.7
Divide by .
Step 3
To remove the radical on the left side of the equation, square both sides of the equation.
Step 4
Step 4.1
Use to rewrite as .
Step 4.2
Simplify the left side.
Step 4.2.1
Simplify .
Step 4.2.1.1
Expand using the FOIL Method.
Step 4.2.1.1.1
Apply the distributive property.
Step 4.2.1.1.2
Apply the distributive property.
Step 4.2.1.1.3
Apply the distributive property.
Step 4.2.1.2
Simplify and combine like terms.
Step 4.2.1.2.1
Simplify each term.
Step 4.2.1.2.1.1
Multiply by .
Step 4.2.1.2.1.2
Multiply by .
Step 4.2.1.2.1.3
Multiply by .
Step 4.2.1.2.1.4
Rewrite using the commutative property of multiplication.
Step 4.2.1.2.1.5
Multiply .
Step 4.2.1.2.1.5.1
Raise to the power of .
Step 4.2.1.2.1.5.2
Raise to the power of .
Step 4.2.1.2.1.5.3
Use the power rule to combine exponents.
Step 4.2.1.2.1.5.4
Add and .
Step 4.2.1.2.2
Add and .
Step 4.2.1.2.3
Add and .
Step 4.2.1.3
Apply pythagorean identity.
Step 4.2.1.4
Multiply the exponents in .
Step 4.2.1.4.1
Apply the power rule and multiply exponents, .
Step 4.2.1.4.2
Cancel the common factor of .
Step 4.2.1.4.2.1
Cancel the common factor.
Step 4.2.1.4.2.2
Rewrite the expression.
Step 4.2.1.5
Simplify.
Step 4.2.1.6
Rewrite in terms of sines and cosines, then cancel the common factors.
Step 4.2.1.6.1
Reorder and .
Step 4.2.1.6.2
Rewrite in terms of sines and cosines.
Step 4.2.1.6.3
Cancel the common factors.
Step 5
Step 5.1
Since the exponents are equal, the bases of the exponents on both sides of the equation must be equal.
Step 5.2
Solve for .
Step 5.2.1
Rewrite the absolute value equation as four equations without absolute value bars.
Step 5.2.2
After simplifying, there are only two unique equations to be solved.
Step 5.2.3
Solve for .
Step 5.2.3.1
For the two functions to be equal, the arguments of each must be equal.
Step 5.2.3.2
Move all terms containing to the left side of the equation.
Step 5.2.3.2.1
Subtract from both sides of the equation.
Step 5.2.3.2.2
Subtract from .
Step 5.2.3.3
Since , the equation will always be true.
All real numbers
All real numbers
Step 5.2.4
Solve for .
Step 5.2.4.1
Move all terms containing to the left side of the equation.
Step 5.2.4.1.1
Add to both sides of the equation.
Step 5.2.4.1.2
Add and .
Step 5.2.4.2
Divide each term in by and simplify.
Step 5.2.4.2.1
Divide each term in by .
Step 5.2.4.2.2
Simplify the left side.
Step 5.2.4.2.2.1
Cancel the common factor of .
Step 5.2.4.2.2.1.1
Cancel the common factor.
Step 5.2.4.2.2.1.2
Divide by .
Step 5.2.4.2.3
Simplify the right side.
Step 5.2.4.2.3.1
Divide by .
Step 5.2.4.3
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 5.2.4.4
Simplify the right side.
Step 5.2.4.4.1
The exact value of is .
Step 5.2.4.5
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 5.2.4.6
Subtract from .
Step 5.2.4.7
Find the period of .
Step 5.2.4.7.1
The period of the function can be calculated using .
Step 5.2.4.7.2
Replace with in the formula for period.
Step 5.2.4.7.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 5.2.4.7.4
Divide by .
Step 5.2.4.8
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 6
Consolidate the answers.
, for any integer