Trigonometry Examples

Solve for x square root of (cos(x))/(tan(x))=cot(x)
Step 1
Subtract from both sides of the equation.
Step 2
Simplify .
Tap for more steps...
Step 2.1
Simplify each term.
Tap for more steps...
Step 2.1.1
Rewrite in terms of sines and cosines.
Step 2.1.2
Multiply by the reciprocal of the fraction to divide by .
Step 2.1.3
Write as a fraction with denominator .
Step 2.1.4
Simplify.
Tap for more steps...
Step 2.1.4.1
Divide by .
Step 2.1.4.2
Combine and .
Step 2.1.5
Simplify the numerator.
Tap for more steps...
Step 2.1.5.1
Raise to the power of .
Step 2.1.5.2
Raise to the power of .
Step 2.1.5.3
Use the power rule to combine exponents.
Step 2.1.5.4
Add and .
Step 2.1.6
Rewrite as .
Step 2.1.7
Pull terms out from under the radical, assuming positive real numbers.
Step 2.1.8
Multiply by .
Step 2.1.9
Combine and simplify the denominator.
Tap for more steps...
Step 2.1.9.1
Multiply by .
Step 2.1.9.2
Raise to the power of .
Step 2.1.9.3
Raise to the power of .
Step 2.1.9.4
Use the power rule to combine exponents.
Step 2.1.9.5
Add and .
Step 2.1.9.6
Rewrite as .
Tap for more steps...
Step 2.1.9.6.1
Use to rewrite as .
Step 2.1.9.6.2
Apply the power rule and multiply exponents, .
Step 2.1.9.6.3
Combine and .
Step 2.1.9.6.4
Cancel the common factor of .
Tap for more steps...
Step 2.1.9.6.4.1
Cancel the common factor.
Step 2.1.9.6.4.2
Rewrite the expression.
Step 2.1.9.6.5
Simplify.
Step 2.1.10
Rewrite in terms of sines and cosines.
Step 2.2
Simplify each term.
Tap for more steps...
Step 2.2.1
Separate fractions.
Step 2.2.2
Convert from to .
Step 2.2.3
Divide by .
Step 2.2.4
Convert from to .
Step 3
Factor .
Tap for more steps...
Step 3.1
Use to rewrite as .
Step 3.2
Factor out of .
Tap for more steps...
Step 3.2.1
Factor out of .
Step 3.2.2
Factor out of .
Step 3.2.3
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
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
Take the inverse cotangent of both sides of the equation to extract from inside the cotangent.
Step 5.2.2
Simplify the right side.
Tap for more steps...
Step 5.2.2.1
The exact value of is .
Step 5.2.3
The cotangent 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 5.2.4
Simplify .
Tap for more steps...
Step 5.2.4.1
To write as a fraction with a common denominator, multiply by .
Step 5.2.4.2
Combine fractions.
Tap for more steps...
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.
Tap for more steps...
Step 5.2.4.3.1
Move to the left of .
Step 5.2.4.3.2
Add and .
Step 5.2.5
Find the period of .
Tap for more steps...
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
Set equal to and solve for .
Tap for more steps...
Step 6.1
Set equal to .
Step 6.2
Solve for .
Tap for more steps...
Step 6.2.1
Add to both sides of the equation.
Step 6.2.2
Raise each side of the equation to the power of to eliminate the fractional exponent on the left side.
Step 6.2.3
Simplify the exponent.
Tap for more steps...
Step 6.2.3.1
Simplify the left side.
Tap for more steps...
Step 6.2.3.1.1
Simplify .
Tap for more steps...
Step 6.2.3.1.1.1
Multiply the exponents in .
Tap for more steps...
Step 6.2.3.1.1.1.1
Apply the power rule and multiply exponents, .
Step 6.2.3.1.1.1.2
Cancel the common factor of .
Tap for more steps...
Step 6.2.3.1.1.1.2.1
Cancel the common factor.
Step 6.2.3.1.1.1.2.2
Rewrite the expression.
Step 6.2.3.1.1.2
Simplify.
Step 6.2.3.2
Simplify the right side.
Tap for more steps...
Step 6.2.3.2.1
One to any power is one.
Step 6.2.4
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 6.2.5
Simplify the right side.
Tap for more steps...
Step 6.2.5.1
The exact value of is .
Step 6.2.6
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 6.2.7
Simplify .
Tap for more steps...
Step 6.2.7.1
To write as a fraction with a common denominator, multiply by .
Step 6.2.7.2
Combine fractions.
Tap for more steps...
Step 6.2.7.2.1
Combine and .
Step 6.2.7.2.2
Combine the numerators over the common denominator.
Step 6.2.7.3
Simplify the numerator.
Tap for more steps...
Step 6.2.7.3.1
Move to the left of .
Step 6.2.7.3.2
Subtract from .
Step 6.2.8
Find the period of .
Tap for more steps...
Step 6.2.8.1
The period of the function can be calculated using .
Step 6.2.8.2
Replace with in the formula for period.
Step 6.2.8.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 6.2.8.4
Divide by .
Step 6.2.9
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 7
The final solution is all the values that make true.
, for any integer
Step 8
Consolidate the answers.
, for any integer