Enter a problem...
Trigonometry Examples
Step 1
Replace the with based on the identity.
Step 2
Subtract from .
Step 3
Reorder the polynomial.
Step 4
Subtract from both sides of the equation.
Step 5
Step 5.1
Divide each term in by .
Step 5.2
Simplify the left side.
Step 5.2.1
Cancel the common factor of .
Step 5.2.1.1
Cancel the common factor.
Step 5.2.1.2
Divide by .
Step 5.3
Simplify the right side.
Step 5.3.1
Dividing two negative values results in a positive value.
Step 6
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 7
Step 7.1
Rewrite as .
Step 7.2
Any root of is .
Step 7.3
Simplify the denominator.
Step 7.3.1
Rewrite as .
Step 7.3.2
Pull terms out from under the radical, assuming positive real numbers.
Step 8
Step 8.1
First, use the positive value of the to find the first solution.
Step 8.2
Next, use the negative value of the to find the second solution.
Step 8.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 9
Set up each of the solutions to solve for .
Step 10
Step 10.1
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 10.2
Simplify the right side.
Step 10.2.1
The exact value of is .
Step 10.3
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 10.4
Simplify .
Step 10.4.1
To write as a fraction with a common denominator, multiply by .
Step 10.4.2
Combine fractions.
Step 10.4.2.1
Combine and .
Step 10.4.2.2
Combine the numerators over the common denominator.
Step 10.4.3
Simplify the numerator.
Step 10.4.3.1
Move to the left of .
Step 10.4.3.2
Subtract from .
Step 10.5
Find the period of .
Step 10.5.1
The period of the function can be calculated using .
Step 10.5.2
Replace with in the formula for period.
Step 10.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 10.5.4
Divide by .
Step 10.6
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
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
The exact value of is .
Step 11.3
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 11.4
Simplify the expression to find the second solution.
Step 11.4.1
Subtract from .
Step 11.4.2
The resulting angle of is positive, less than , and coterminal with .
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
Add to every negative angle to get positive angles.
Step 11.6.1
Add to to find the positive angle.
Step 11.6.2
To write as a fraction with a common denominator, multiply by .
Step 11.6.3
Combine fractions.
Step 11.6.3.1
Combine and .
Step 11.6.3.2
Combine the numerators over the common denominator.
Step 11.6.4
Simplify the numerator.
Step 11.6.4.1
Multiply by .
Step 11.6.4.2
Subtract from .
Step 11.6.5
List the new angles.
Step 11.7
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Step 12
List all of the solutions.
, for any integer
Step 13
Step 13.1
Consolidate and to .
, for any integer
Step 13.2
Consolidate and to .
, for any integer
, for any integer