Trigonometry Examples

Solve for x sin(x/2)=cos(x/2)
Step 1
Divide each term in the equation by .
Step 2
Convert from to .
Step 3
Cancel the common factor of .
Tap for more steps...
Step 3.1
Cancel the common factor.
Step 3.2
Rewrite the expression.
Step 4
Take the inverse tangent of both sides of the equation to extract from inside the tangent.
Step 5
Simplify the right side.
Tap for more steps...
Step 5.1
The exact value of is .
Step 6
Multiply both sides of the equation by .
Step 7
Simplify both sides of the equation.
Tap for more steps...
Step 7.1
Simplify the left side.
Tap for more steps...
Step 7.1.1
Cancel the common factor of .
Tap for more steps...
Step 7.1.1.1
Cancel the common factor.
Step 7.1.1.2
Rewrite the expression.
Step 7.2
Simplify the right side.
Tap for more steps...
Step 7.2.1
Cancel the common factor of .
Tap for more steps...
Step 7.2.1.1
Factor out of .
Step 7.2.1.2
Cancel the common factor.
Step 7.2.1.3
Rewrite the expression.
Step 8
The tangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from to find the solution in the fourth quadrant.
Step 9
Solve for .
Tap for more steps...
Step 9.1
Multiply both sides of the equation by .
Step 9.2
Simplify both sides of the equation.
Tap for more steps...
Step 9.2.1
Simplify the left side.
Tap for more steps...
Step 9.2.1.1
Cancel the common factor of .
Tap for more steps...
Step 9.2.1.1.1
Cancel the common factor.
Step 9.2.1.1.2
Rewrite the expression.
Step 9.2.2
Simplify the right side.
Tap for more steps...
Step 9.2.2.1
Simplify .
Tap for more steps...
Step 9.2.2.1.1
To write as a fraction with a common denominator, multiply by .
Step 9.2.2.1.2
Simplify terms.
Tap for more steps...
Step 9.2.2.1.2.1
Combine and .
Step 9.2.2.1.2.2
Combine the numerators over the common denominator.
Step 9.2.2.1.2.3
Cancel the common factor of .
Tap for more steps...
Step 9.2.2.1.2.3.1
Factor out of .
Step 9.2.2.1.2.3.2
Cancel the common factor.
Step 9.2.2.1.2.3.3
Rewrite the expression.
Step 9.2.2.1.3
Simplify the numerator.
Tap for more steps...
Step 9.2.2.1.3.1
Move to the left of .
Step 9.2.2.1.3.2
Add and .
Step 10
Find the period of .
Tap for more steps...
Step 10.1
The period of the function can be calculated using .
Step 10.2
Replace with in the formula for period.
Step 10.3
is approximately which is positive so remove the absolute value
Step 10.4
Multiply the numerator by the reciprocal of the denominator.
Step 10.5
Move to the left of .
Step 11
The period of the function is so values will repeat every radians in both directions.
, for any integer
Step 12
Consolidate the answers.
, for any integer