Trigonometry Examples

Solve for x 3sin(x)=2cos(x)^2
Step 1
Subtract from both sides of the equation.
Step 2
Replace the with based on the identity.
Step 3
Simplify each term.
Tap for more steps...
Step 3.1
Apply the distributive property.
Step 3.2
Multiply by .
Step 3.3
Multiply by .
Step 4
Reorder the polynomial.
Step 5
Substitute for .
Step 6
Factor by grouping.
Tap for more steps...
Step 6.1
For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .
Tap for more steps...
Step 6.1.1
Factor out of .
Step 6.1.2
Rewrite as plus
Step 6.1.3
Apply the distributive property.
Step 6.2
Factor out the greatest common factor from each group.
Tap for more steps...
Step 6.2.1
Group the first two terms and the last two terms.
Step 6.2.2
Factor out the greatest common factor (GCF) from each group.
Step 6.3
Factor the polynomial by factoring out the greatest common factor, .
Step 7
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 8
Set equal to and solve for .
Tap for more steps...
Step 8.1
Set equal to .
Step 8.2
Solve for .
Tap for more steps...
Step 8.2.1
Add to both sides of the equation.
Step 8.2.2
Divide each term in by and simplify.
Tap for more steps...
Step 8.2.2.1
Divide each term in by .
Step 8.2.2.2
Simplify the left side.
Tap for more steps...
Step 8.2.2.2.1
Cancel the common factor of .
Tap for more steps...
Step 8.2.2.2.1.1
Cancel the common factor.
Step 8.2.2.2.1.2
Divide by .
Step 9
Set equal to and solve for .
Tap for more steps...
Step 9.1
Set equal to .
Step 9.2
Subtract from both sides of the equation.
Step 10
The final solution is all the values that make true.
Step 11
Substitute for .
Step 12
Set up each of the solutions to solve for .
Step 13
Solve for in .
Tap for more steps...
Step 13.1
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 13.2
Simplify the right side.
Tap for more steps...
Step 13.2.1
The exact value of is .
Step 13.3
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 13.4
Simplify .
Tap for more steps...
Step 13.4.1
To write as a fraction with a common denominator, multiply by .
Step 13.4.2
Combine fractions.
Tap for more steps...
Step 13.4.2.1
Combine and .
Step 13.4.2.2
Combine the numerators over the common denominator.
Step 13.4.3
Simplify the numerator.
Tap for more steps...
Step 13.4.3.1
Move to the left of .
Step 13.4.3.2
Subtract from .
Step 13.5
Find the period of .
Tap for more steps...
Step 13.5.1
The period of the function can be calculated using .
Step 13.5.2
Replace with in the formula for period.
Step 13.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 13.5.4
Divide by .
Step 13.6
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Step 14
Solve for in .
Tap for more steps...
Step 14.1
The range of sine is . Since does not fall in this range, there is no solution.
No solution
No solution
Step 15
List all of the solutions.
, for any integer