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Trigonometry Examples
Step 1
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 2
Step 2.1
The exact value of is .
Step 3
Step 3.1
Subtract from both sides of the equation.
Step 3.2
Combine the numerators over the common denominator.
Step 3.3
Subtract from .
Step 3.4
Divide by .
Step 4
Step 4.1
Divide each term in by .
Step 4.2
Simplify the left side.
Step 4.2.1
Cancel the common factor of .
Step 4.2.1.1
Cancel the common factor.
Step 4.2.1.2
Divide by .
Step 4.3
Simplify the right side.
Step 4.3.1
Divide by .
Step 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 6
Step 6.1
Simplify .
Step 6.1.1
To write as a fraction with a common denominator, multiply by .
Step 6.1.2
Combine fractions.
Step 6.1.2.1
Combine and .
Step 6.1.2.2
Combine the numerators over the common denominator.
Step 6.1.3
Simplify the numerator.
Step 6.1.3.1
Move to the left of .
Step 6.1.3.2
Subtract from .
Step 6.2
Move all terms not containing to the right side of the equation.
Step 6.2.1
Subtract from both sides of the equation.
Step 6.2.2
Combine the numerators over the common denominator.
Step 6.2.3
Subtract from .
Step 6.2.4
Cancel the common factor of and .
Step 6.2.4.1
Factor out of .
Step 6.2.4.2
Cancel the common factors.
Step 6.2.4.2.1
Factor out of .
Step 6.2.4.2.2
Cancel the common factor.
Step 6.2.4.2.3
Rewrite the expression.
Step 6.3
Divide each term in by and simplify.
Step 6.3.1
Divide each term in by .
Step 6.3.2
Simplify the left side.
Step 6.3.2.1
Cancel the common factor of .
Step 6.3.2.1.1
Cancel the common factor.
Step 6.3.2.1.2
Divide by .
Step 6.3.3
Simplify the right side.
Step 6.3.3.1
Multiply the numerator by the reciprocal of the denominator.
Step 6.3.3.2
Cancel the common factor of .
Step 6.3.3.2.1
Factor out of .
Step 6.3.3.2.2
Cancel the common factor.
Step 6.3.3.2.3
Rewrite the expression.
Step 7
Step 7.1
The period of the function can be calculated using .
Step 7.2
Replace with in the formula for period.
Step 7.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 7.4
Cancel the common factor of .
Step 7.4.1
Cancel the common factor.
Step 7.4.2
Divide by .
Step 8
The period of the function is so values will repeat every radians in both directions.
, for any integer