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Trigonometry Examples
Step 1
Subtract from both sides of the equation.
Step 2
Step 2.1
Divide each term in by .
Step 2.2
Simplify the left side.
Step 2.2.1
Cancel the common factor of .
Step 2.2.1.1
Cancel the common factor.
Step 2.2.1.2
Divide by .
Step 2.3
Simplify the right side.
Step 2.3.1
Move the negative in front of the fraction.
Step 3
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 4
Step 4.1
The exact value of is .
Step 5
Multiply both sides of the equation by .
Step 6
Step 6.1
Simplify the left side.
Step 6.1.1
Cancel the common factor of .
Step 6.1.1.1
Cancel the common factor.
Step 6.1.1.2
Rewrite the expression.
Step 6.2
Simplify the right side.
Step 6.2.1
Simplify .
Step 6.2.1.1
Multiply .
Step 6.2.1.1.1
Multiply by .
Step 6.2.1.1.2
Combine and .
Step 6.2.1.2
Move the negative in front of the fraction.
Step 7
The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from , to find a reference angle. Next, add this reference angle to to find the solution in the third quadrant.
Step 8
Step 8.1
Subtract from .
Step 8.2
The resulting angle of is positive, less than , and coterminal with .
Step 8.3
Solve for .
Step 8.3.1
Multiply both sides of the equation by .
Step 8.3.2
Simplify both sides of the equation.
Step 8.3.2.1
Simplify the left side.
Step 8.3.2.1.1
Cancel the common factor of .
Step 8.3.2.1.1.1
Cancel the common factor.
Step 8.3.2.1.1.2
Rewrite the expression.
Step 8.3.2.2
Simplify the right side.
Step 8.3.2.2.1
Multiply .
Step 8.3.2.2.1.1
Combine and .
Step 8.3.2.2.1.2
Multiply by .
Step 9
Step 9.1
The period of the function can be calculated using .
Step 9.2
Replace with in the formula for period.
Step 9.3
is approximately which is positive so remove the absolute value
Step 9.4
Multiply the numerator by the reciprocal of the denominator.
Step 9.5
Multiply by .
Step 10
Step 10.1
Add to to find the positive angle.
Step 10.2
To write as a fraction with a common denominator, multiply by .
Step 10.3
Combine fractions.
Step 10.3.1
Combine and .
Step 10.3.2
Combine the numerators over the common denominator.
Step 10.4
Simplify the numerator.
Step 10.4.1
Multiply by .
Step 10.4.2
Subtract from .
Step 10.5
List the new angles.
Step 11
The period of the function is so values will repeat every radians in both directions.
, for any integer