Trigonometry Examples

Solve for x sin(x)cot(x)+1/( square root of 2)=0
Step 1
Simplify the left side.
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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, then cancel the common factors.
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Step 1.1.1.1
Reorder and .
Step 1.1.1.2
Rewrite in terms of sines and cosines.
Step 1.1.1.3
Cancel the common factors.
Step 1.1.2
Multiply by .
Step 1.1.3
Combine and simplify the denominator.
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Step 1.1.3.1
Multiply by .
Step 1.1.3.2
Raise to the power of .
Step 1.1.3.3
Raise to the power of .
Step 1.1.3.4
Use the power rule to combine exponents.
Step 1.1.3.5
Add and .
Step 1.1.3.6
Rewrite as .
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Step 1.1.3.6.1
Use to rewrite as .
Step 1.1.3.6.2
Apply the power rule and multiply exponents, .
Step 1.1.3.6.3
Combine and .
Step 1.1.3.6.4
Cancel the common factor of .
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Step 1.1.3.6.4.1
Cancel the common factor.
Step 1.1.3.6.4.2
Rewrite the expression.
Step 1.1.3.6.5
Evaluate the exponent.
Step 2
Subtract from both sides of the equation.
Step 3
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 4
Simplify the right side.
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Step 4.1
The exact value of is .
Step 5
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 6
Simplify .
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Step 6.1
To write as a fraction with a common denominator, multiply by .
Step 6.2
Combine fractions.
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Step 6.2.1
Combine and .
Step 6.2.2
Combine the numerators over the common denominator.
Step 6.3
Simplify the numerator.
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Step 6.3.1
Multiply by .
Step 6.3.2
Subtract from .
Step 7
Find the period of .
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Step 7.1
The period of the function can be calculated using .
Step 7.2
Replace with in the formula for period.
Step 7.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 7.4
Divide by .
Step 8
The period of the function is so values will repeat every radians in both directions.
, for any integer