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Trigonometry Examples
Step 1
Step 1.1
Simplify .
Step 1.1.1
Combine and .
Step 1.1.2
Move to the left of .
Step 2
Step 2.1
Divide each term in by .
Step 2.2
Simplify the left side.
Step 2.2.1
Cancel the common factor of .
Step 2.2.1.1
Cancel the common factor.
Step 2.2.1.2
Rewrite the expression.
Step 2.3
Simplify the right side.
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.
Step 2.3.5.1
Rewrite the expression.
Step 2.3.5.2
Multiply by .
Step 2.3.6
Cancel the common factor of .
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
Step 5.1
Simplify the left side.
Step 5.1.1
Simplify .
Step 5.1.1.1
Cancel the common factor of .
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 .
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.
Step 5.1.1.3.1
Multiply by .
Step 5.1.1.3.2
Multiply by .
Step 5.2
Simplify the right side.
Step 5.2.1
Simplify .
Step 5.2.1.1
Multiply by .
Step 5.2.1.2
Combine and simplify the denominator.
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 .
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 .
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 .
Step 5.2.1.4.1
Factor out of .
Step 5.2.1.4.2
Cancel the common factors.
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
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
Step 9.1
To write as a fraction with a common denominator, multiply by .
Step 9.2
Combine fractions.
Step 9.2.1
Combine and .
Step 9.2.2
Combine the numerators over the common denominator.
Step 9.3
Simplify the numerator.
Step 9.3.1
Multiply by .
Step 9.3.2
Subtract from .
Step 10
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