Trigonometry Examples

Solve for x sin(x)^2=2sin(x)+3
Step 1
Move all the expressions to the left side of the equation.
Tap for more steps...
Step 1.1
Subtract from both sides of the equation.
Step 1.2
Subtract from both sides of the equation.
Step 2
Factor using the AC method.
Tap for more steps...
Step 2.1
Consider the form . Find a pair of integers whose product is and whose sum is . In this case, whose product is and whose sum is .
Step 2.2
Write the factored form using these integers.
Step 3
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 4
Set equal to and solve for .
Tap for more steps...
Step 4.1
Set equal to .
Step 4.2
Solve for .
Tap for more steps...
Step 4.2.1
Add to both sides of the equation.
Step 4.2.2
The range of sine is . Since does not fall in this range, there is no solution.
No solution
No solution
No solution
Step 5
Set equal to and solve for .
Tap for more steps...
Step 5.1
Set equal to .
Step 5.2
Solve for .
Tap for more steps...
Step 5.2.1
Subtract from both sides of the equation.
Step 5.2.2
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 5.2.3
Simplify the right side.
Tap for more steps...
Step 5.2.3.1
The exact value of is .
Step 5.2.4
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 5.2.5
Simplify the expression to find the second solution.
Tap for more steps...
Step 5.2.5.1
Subtract from .
Step 5.2.5.2
The resulting angle of is positive, less than , and coterminal with .
Step 5.2.6
Find the period of .
Tap for more steps...
Step 5.2.6.1
The period of the function can be calculated using .
Step 5.2.6.2
Replace with in the formula for period.
Step 5.2.6.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 5.2.6.4
Divide by .
Step 5.2.7
Add to every negative angle to get positive angles.
Tap for more steps...
Step 5.2.7.1
Add to to find the positive angle.
Step 5.2.7.2
To write as a fraction with a common denominator, multiply by .
Step 5.2.7.3
Combine fractions.
Tap for more steps...
Step 5.2.7.3.1
Combine and .
Step 5.2.7.3.2
Combine the numerators over the common denominator.
Step 5.2.7.4
Simplify the numerator.
Tap for more steps...
Step 5.2.7.4.1
Multiply by .
Step 5.2.7.4.2
Subtract from .
Step 5.2.7.5
List the new angles.
Step 5.2.8
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
The final solution is all the values that make true.
, for any integer