Trigonometry Examples

Solve for ? tan(x)=-(2 square root of 3)/3*sin(x)
Step 1
Simplify the right side.
Tap for more steps...
Step 1.1
Simplify .
Tap for more steps...
Step 1.1.1
Combine and .
Step 1.1.2
Move to the left of .
Step 2
Divide each term in by and simplify.
Tap for more steps...
Step 2.1
Divide each term in by .
Step 2.2
Simplify the left side.
Tap for more steps...
Step 2.2.1
Cancel the common factor of .
Tap for more steps...
Step 2.2.1.1
Cancel the common factor.
Step 2.2.1.2
Rewrite the expression.
Step 2.3
Simplify the right side.
Tap for more steps...
Step 2.3.1
Multiply the numerator by the reciprocal of the denominator.
Step 2.3.2
Rewrite in terms of sines and cosines.
Step 2.3.3
Multiply by the reciprocal of the fraction to divide by .
Step 2.3.4
Write as a fraction with denominator .
Step 2.3.5
Simplify.
Tap for more steps...
Step 2.3.5.1
Rewrite the expression.
Step 2.3.5.2
Multiply by .
Step 2.3.6
Cancel the common factor of .
Tap for more steps...
Step 2.3.6.1
Move the leading negative in into the numerator.
Step 2.3.6.2
Factor out of .
Step 2.3.6.3
Cancel the common factor.
Step 2.3.6.4
Rewrite the expression.
Step 2.3.7
Combine and .
Step 2.3.8
Move the negative in front of the fraction.
Step 3
Rewrite the equation as .
Step 4
Multiply both sides of the equation by .
Step 5
Simplify both sides of the equation.
Tap for more steps...
Step 5.1
Simplify the left side.
Tap for more steps...
Step 5.1.1
Simplify .
Tap for more steps...
Step 5.1.1.1
Cancel the common factor of .
Tap for more steps...
Step 5.1.1.1.1
Move the leading negative in into the numerator.
Step 5.1.1.1.2
Move the leading negative in into the numerator.
Step 5.1.1.1.3
Factor out of .
Step 5.1.1.1.4
Cancel the common factor.
Step 5.1.1.1.5
Rewrite the expression.
Step 5.1.1.2
Cancel the common factor of .
Tap for more steps...
Step 5.1.1.2.1
Factor out of .
Step 5.1.1.2.2
Cancel the common factor.
Step 5.1.1.2.3
Rewrite the expression.
Step 5.1.1.3
Multiply.
Tap for more steps...
Step 5.1.1.3.1
Multiply by .
Step 5.1.1.3.2
Multiply by .
Step 5.2
Simplify the right side.
Tap for more steps...
Step 5.2.1
Simplify .
Tap for more steps...
Step 5.2.1.1
Multiply by .
Step 5.2.1.2
Combine and simplify the denominator.
Tap for more steps...
Step 5.2.1.2.1
Multiply by .
Step 5.2.1.2.2
Move .
Step 5.2.1.2.3
Raise to the power of .
Step 5.2.1.2.4
Raise to the power of .
Step 5.2.1.2.5
Use the power rule to combine exponents.
Step 5.2.1.2.6
Add and .
Step 5.2.1.2.7
Rewrite as .
Tap for more steps...
Step 5.2.1.2.7.1
Use to rewrite as .
Step 5.2.1.2.7.2
Apply the power rule and multiply exponents, .
Step 5.2.1.2.7.3
Combine and .
Step 5.2.1.2.7.4
Cancel the common factor of .
Tap for more steps...
Step 5.2.1.2.7.4.1
Cancel the common factor.
Step 5.2.1.2.7.4.2
Rewrite the expression.
Step 5.2.1.2.7.5
Evaluate the exponent.
Step 5.2.1.3
Multiply by .
Step 5.2.1.4
Cancel the common factor of and .
Tap for more steps...
Step 5.2.1.4.1
Factor out of .
Step 5.2.1.4.2
Cancel the common factors.
Tap for more steps...
Step 5.2.1.4.2.1
Factor out of .
Step 5.2.1.4.2.2
Cancel the common factor.
Step 5.2.1.4.2.3
Rewrite the expression.
Step 5.2.1.5
Multiply by .
Step 6
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 7
Simplify the right side.
Tap for more steps...
Step 7.1
The exact value of is .
Step 8
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 9
Simplify .
Tap for more steps...
Step 9.1
To write as a fraction with a common denominator, multiply by .
Step 9.2
Combine fractions.
Tap for more steps...
Step 9.2.1
Combine and .
Step 9.2.2
Combine the numerators over the common denominator.
Step 9.3
Simplify the numerator.
Tap for more steps...
Step 9.3.1
Multiply by .
Step 9.3.2
Subtract from .
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
The absolute value is the distance between a number and zero. The distance between and is .
Step 10.4
Divide by .
Step 11
The period of the function is so values will repeat every radians in both directions.
, for any integer