Trigonometry Examples

Solve for s cos(s)^2=1/4
Step 1
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 2
Simplify .
Tap for more steps...
Step 2.1
Rewrite as .
Step 2.2
Any root of is .
Step 2.3
Simplify the denominator.
Tap for more steps...
Step 2.3.1
Rewrite as .
Step 2.3.2
Pull terms out from under the radical, assuming positive real numbers.
Step 3
The complete solution is the result of both the positive and negative portions of the solution.
Tap for more steps...
Step 3.1
First, use the positive value of the to find the first solution.
Step 3.2
Next, use the negative value of the to find the second solution.
Step 3.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 4
Set up each of the solutions to solve for .
Step 5
Solve for in .
Tap for more steps...
Step 5.1
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 5.2
Simplify the right side.
Tap for more steps...
Step 5.2.1
The exact value of is .
Step 5.3
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 5.4
Simplify .
Tap for more steps...
Step 5.4.1
To write as a fraction with a common denominator, multiply by .
Step 5.4.2
Combine fractions.
Tap for more steps...
Step 5.4.2.1
Combine and .
Step 5.4.2.2
Combine the numerators over the common denominator.
Step 5.4.3
Simplify the numerator.
Tap for more steps...
Step 5.4.3.1
Multiply by .
Step 5.4.3.2
Subtract from .
Step 5.5
Find the period of .
Tap for more steps...
Step 5.5.1
The period of the function can be calculated using .
Step 5.5.2
Replace with in the formula for period.
Step 5.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 5.5.4
Divide by .
Step 5.6
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Step 6
Solve for in .
Tap for more steps...
Step 6.1
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 6.2
Simplify the right side.
Tap for more steps...
Step 6.2.1
The exact value of is .
Step 6.3
The cosine function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from to find the solution in the third quadrant.
Step 6.4
Simplify .
Tap for more steps...
Step 6.4.1
To write as a fraction with a common denominator, multiply by .
Step 6.4.2
Combine fractions.
Tap for more steps...
Step 6.4.2.1
Combine and .
Step 6.4.2.2
Combine the numerators over the common denominator.
Step 6.4.3
Simplify the numerator.
Tap for more steps...
Step 6.4.3.1
Multiply by .
Step 6.4.3.2
Subtract from .
Step 6.5
Find the period of .
Tap for more steps...
Step 6.5.1
The period of the function can be calculated using .
Step 6.5.2
Replace with in the formula for period.
Step 6.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 6.5.4
Divide by .
Step 6.6
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Step 7
List all of the solutions.
, for any integer
Step 8
Consolidate the solutions.
Tap for more steps...
Step 8.1
Consolidate and to .
, for any integer
Step 8.2
Consolidate and to .
, for any integer
, for any integer