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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
Since the expression on each side of the equation has the same denominator, the numerators must be equal.
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 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
Multiply both sides of the equation by .
Step 6.2
Simplify both sides of the equation.
Step 6.2.1
Simplify the left side.
Step 6.2.1.1
Simplify .
Step 6.2.1.1.1
Cancel the common factor of .
Step 6.2.1.1.1.1
Cancel the common factor.
Step 6.2.1.1.1.2
Rewrite the expression.
Step 6.2.1.1.2
Cancel the common factor of .
Step 6.2.1.1.2.1
Factor out of .
Step 6.2.1.1.2.2
Cancel the common factor.
Step 6.2.1.1.2.3
Rewrite the expression.
Step 6.2.2
Simplify the right side.
Step 6.2.2.1
Simplify .
Step 6.2.2.1.1
To write as a fraction with a common denominator, multiply by .
Step 6.2.2.1.2
Simplify terms.
Step 6.2.2.1.2.1
Combine and .
Step 6.2.2.1.2.2
Combine the numerators over the common denominator.
Step 6.2.2.1.2.3
Cancel the common factor of .
Step 6.2.2.1.2.3.1
Cancel the common factor.
Step 6.2.2.1.2.3.2
Rewrite the expression.
Step 6.2.2.1.3
Move to the left of .
Step 6.2.2.1.4
Simplify terms.
Step 6.2.2.1.4.1
Subtract from .
Step 6.2.2.1.4.2
Combine and .
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
is approximately which is positive so remove the absolute value
Step 7.4
Multiply the numerator by the reciprocal of the denominator.
Step 7.5
Multiply .
Step 7.5.1
Combine and .
Step 7.5.2
Multiply by .
Step 7.5.3
Combine and .
Step 8
The period of the function is so values will repeat every radians in both directions.
, for any integer