Trigonometry Examples

Solve for ? 8sin(x)^2tan(x)-8sin(x)^2=0
Step 1
Simplify the left side of the equation.
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Step 1.1
Simplify each term.
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Step 1.1.1
Rewrite in terms of sines and cosines.
Step 1.1.2
Multiply .
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Step 1.1.2.1
Combine and .
Step 1.1.2.2
Combine and .
Step 1.1.2.3
Multiply by by adding the exponents.
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Step 1.1.2.3.1
Move .
Step 1.1.2.3.2
Multiply by .
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Step 1.1.2.3.2.1
Raise to the power of .
Step 1.1.2.3.2.2
Use the power rule to combine exponents.
Step 1.1.2.3.3
Add and .
Step 1.1.3
Move to the left of .
Step 1.2
Simplify each term.
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Step 1.2.1
Factor out of .
Step 1.2.2
Separate fractions.
Step 1.2.3
Convert from to .
Step 1.2.4
Divide by .
Step 2
Factor out of .
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Step 2.1
Factor out of .
Step 2.2
Factor out of .
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 .
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Step 4.1
Set equal to .
Step 4.2
Solve for .
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Step 4.2.1
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 4.2.2
Simplify .
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Step 4.2.2.1
Rewrite as .
Step 4.2.2.2
Pull terms out from under the radical, assuming positive real numbers.
Step 4.2.2.3
Plus or minus is .
Step 4.2.3
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 4.2.4
Simplify the right side.
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Step 4.2.4.1
The exact value of is .
Step 4.2.5
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 4.2.6
Subtract from .
Step 4.2.7
Find the period of .
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Step 4.2.7.1
The period of the function can be calculated using .
Step 4.2.7.2
Replace with in the formula for period.
Step 4.2.7.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 4.2.7.4
Divide by .
Step 4.2.8
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 5
Set equal to and solve for .
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Step 5.1
Set equal to .
Step 5.2
Solve for .
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Step 5.2.1
Add to both sides of the equation.
Step 5.2.2
Take the inverse tangent of both sides of the equation to extract from inside the tangent.
Step 5.2.3
Simplify the right side.
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Step 5.2.3.1
The exact value of is .
Step 5.2.4
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 5.2.5
Simplify .
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Step 5.2.5.1
To write as a fraction with a common denominator, multiply by .
Step 5.2.5.2
Combine fractions.
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Step 5.2.5.2.1
Combine and .
Step 5.2.5.2.2
Combine the numerators over the common denominator.
Step 5.2.5.3
Simplify the numerator.
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Step 5.2.5.3.1
Move to the left of .
Step 5.2.5.3.2
Add and .
Step 5.2.6
Find the period of .
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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
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
Step 7
Consolidate the answers.
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Step 7.1
Consolidate and to .
, for any integer
Step 7.2
Consolidate and to .
, for any integer
, for any integer