Enter a problem...
Trigonometry Examples
Step 1
Substitute for .
Step 2
Subtract from both sides of the equation.
Step 3
Step 3.1
Factor out of .
Step 3.2
Factor out of .
Step 3.3
Factor out of .
Step 3.4
Factor out of .
Step 3.5
Factor out of .
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
Use the quadratic formula to find the solutions.
Step 6
Substitute the values , , and into the quadratic formula and solve for .
Step 7
Step 7.1
Simplify the numerator.
Step 7.1.1
Raise to the power of .
Step 7.1.2
Multiply .
Step 7.1.2.1
Multiply by .
Step 7.1.2.2
Multiply by .
Step 7.1.3
Add and .
Step 7.1.4
Rewrite as .
Step 7.1.4.1
Factor out of .
Step 7.1.4.2
Rewrite as .
Step 7.1.5
Pull terms out from under the radical.
Step 7.2
Multiply by .
Step 7.3
Simplify .
Step 8
The final answer is the combination of both solutions.
Step 9
Substitute for .
Step 10
Set up each of the solutions to solve for .
Step 11
Step 11.1
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 11.2
Simplify the right side.
Step 11.2.1
Evaluate .
Step 11.3
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 11.4
Simplify .
Step 11.4.1
To write as a fraction with a common denominator, multiply by .
Step 11.4.2
Combine fractions.
Step 11.4.2.1
Combine and .
Step 11.4.2.2
Combine the numerators over the common denominator.
Step 11.4.3
Simplify the numerator.
Step 11.4.3.1
Move to the left of .
Step 11.4.3.2
Subtract from .
Step 11.5
Find the period of .
Step 11.5.1
The period of the function can be calculated using .
Step 11.5.2
Replace with in the formula for period.
Step 11.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 11.5.4
Divide by .
Step 11.6
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Step 12
Step 12.1
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 12.2
Simplify the right side.
Step 12.2.1
Evaluate .
Step 12.3
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 12.4
Simplify .
Step 12.4.1
To write as a fraction with a common denominator, multiply by .
Step 12.4.2
Combine fractions.
Step 12.4.2.1
Combine and .
Step 12.4.2.2
Combine the numerators over the common denominator.
Step 12.4.3
Simplify the numerator.
Step 12.4.3.1
Move to the left of .
Step 12.4.3.2
Add and .
Step 12.5
Find the period of .
Step 12.5.1
The period of the function can be calculated using .
Step 12.5.2
Replace with in the formula for period.
Step 12.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 12.5.4
Divide by .
Step 12.6
Add to every negative angle to get positive angles.
Step 12.6.1
Add to to find the positive angle.
Step 12.6.2
To write as a fraction with a common denominator, multiply by .
Step 12.6.3
Combine fractions.
Step 12.6.3.1
Combine and .
Step 12.6.3.2
Combine the numerators over the common denominator.
Step 12.6.4
Simplify the numerator.
Step 12.6.4.1
Multiply by .
Step 12.6.4.2
Subtract from .
Step 12.6.5
List the new angles.
Step 12.7
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Step 13
List all of the solutions.
, for any integer