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Trigonometry Examples
Step 1
Subtract from both sides of the equation.
Step 2
Step 2.1
Apply the sine double-angle identity.
Step 2.2
Use the triple-angle identity to transform to .
Step 2.3
Apply the distributive property.
Step 2.4
Multiply by .
Step 2.5
Multiply by .
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
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 5
Step 5.1
Set equal to .
Step 5.2
Solve for .
Step 5.2.1
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 5.2.2
Simplify the right side.
Step 5.2.2.1
The exact value of is .
Step 5.2.3
The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from to find the solution in the fourth quadrant.
Step 5.2.4
Simplify .
Step 5.2.4.1
To write as a fraction with a common denominator, multiply by .
Step 5.2.4.2
Combine fractions.
Step 5.2.4.2.1
Combine and .
Step 5.2.4.2.2
Combine the numerators over the common denominator.
Step 5.2.4.3
Simplify the numerator.
Step 5.2.4.3.1
Multiply by .
Step 5.2.4.3.2
Subtract from .
Step 5.2.5
Find the period of .
Step 5.2.5.1
The period of the function can be calculated using .
Step 5.2.5.2
Replace with in the formula for period.
Step 5.2.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 5.2.5.4
Divide by .
Step 5.2.6
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
, for any integer
Step 6
Step 6.1
Set equal to .
Step 6.2
Solve for .
Step 6.2.1
Replace the with based on the identity.
Step 6.2.2
Simplify each term.
Step 6.2.2.1
Apply the distributive property.
Step 6.2.2.2
Multiply by .
Step 6.2.2.3
Multiply by .
Step 6.2.3
Add and .
Step 6.2.4
Reorder the polynomial.
Step 6.2.5
Substitute for .
Step 6.2.6
Use the quadratic formula to find the solutions.
Step 6.2.7
Substitute the values , , and into the quadratic formula and solve for .
Step 6.2.8
Simplify.
Step 6.2.8.1
Simplify the numerator.
Step 6.2.8.1.1
Raise to the power of .
Step 6.2.8.1.2
Multiply .
Step 6.2.8.1.2.1
Multiply by .
Step 6.2.8.1.2.2
Multiply by .
Step 6.2.8.1.3
Add and .
Step 6.2.8.1.4
Rewrite as .
Step 6.2.8.1.4.1
Factor out of .
Step 6.2.8.1.4.2
Rewrite as .
Step 6.2.8.1.5
Pull terms out from under the radical.
Step 6.2.8.2
Multiply by .
Step 6.2.8.3
Simplify .
Step 6.2.9
The final answer is the combination of both solutions.
Step 6.2.10
Substitute for .
Step 6.2.11
Set up each of the solutions to solve for .
Step 6.2.12
Solve for in .
Step 6.2.12.1
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 6.2.12.2
Simplify the right side.
Step 6.2.12.2.1
Evaluate .
Step 6.2.12.3
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 6.2.12.4
Simplify the expression to find the second solution.
Step 6.2.12.4.1
Subtract from .
Step 6.2.12.4.2
The resulting angle of is positive, less than , and coterminal with .
Step 6.2.12.5
Find the period of .
Step 6.2.12.5.1
The period of the function can be calculated using .
Step 6.2.12.5.2
Replace with in the formula for period.
Step 6.2.12.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 6.2.12.5.4
Divide by .
Step 6.2.12.6
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Step 6.2.13
Solve for in .
Step 6.2.13.1
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 6.2.13.2
Simplify the right side.
Step 6.2.13.2.1
Evaluate .
Step 6.2.13.3
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 6.2.13.4
Simplify the expression to find the second solution.
Step 6.2.13.4.1
Subtract from .
Step 6.2.13.4.2
The resulting angle of is positive, less than , and coterminal with .
Step 6.2.13.5
Find the period of .
Step 6.2.13.5.1
The period of the function can be calculated using .
Step 6.2.13.5.2
Replace with in the formula for period.
Step 6.2.13.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 6.2.13.5.4
Divide by .
Step 6.2.13.6
Add to every negative angle to get positive angles.
Step 6.2.13.6.1
Add to to find the positive angle.
Step 6.2.13.6.2
To write as a fraction with a common denominator, multiply by .
Step 6.2.13.6.3
Combine fractions.
Step 6.2.13.6.3.1
Combine and .
Step 6.2.13.6.3.2
Combine the numerators over the common denominator.
Step 6.2.13.6.4
Simplify the numerator.
Step 6.2.13.6.4.1
Multiply by .
Step 6.2.13.6.4.2
Subtract from .
Step 6.2.13.6.5
List the new angles.
Step 6.2.13.7
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Step 6.2.14
List all of the solutions.
, for any integer
, for any integer
, for any integer
Step 7
The final solution is all the values that make true.
, for any integer
Step 8
Consolidate and to .
, for any integer