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Trigonometry Examples
Step 1
Square both sides of the equation.
Step 2
Step 2.1
Rewrite as .
Step 2.2
Expand using the FOIL Method.
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.
Step 2.3.1
Simplify each term.
Step 2.3.1.1
Multiply .
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 .
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.
Step 2.6.1
Reorder and .
Step 2.6.2
Reorder and .
Step 2.6.3
Apply the sine double-angle identity.
Step 3
Step 3.1
Simplify the expression.
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 .
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 .
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
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
Step 6.1
The exact value of is .
Step 7
Step 7.1
Divide each term in by .
Step 7.2
Simplify the left side.
Step 7.2.1
Cancel the common factor of .
Step 7.2.1.1
Cancel the common factor.
Step 7.2.1.2
Divide by .
Step 7.3
Simplify the right side.
Step 7.3.1
Multiply the numerator by the reciprocal of the denominator.
Step 7.3.2
Multiply .
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
Step 9.1
Simplify.
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 .
Step 9.1.4.1
Reorder and .
Step 9.1.4.2
Subtract from .
Step 9.2
Divide each term in by and simplify.
Step 9.2.1
Divide each term in by .
Step 9.2.2
Simplify the left side.
Step 9.2.2.1
Cancel the common factor of .
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.
Step 9.2.3.1
Multiply the numerator by the reciprocal of the denominator.
Step 9.2.3.2
Multiply .
Step 9.2.3.2.1
Multiply by .
Step 9.2.3.2.2
Multiply by .
Step 10
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 .
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