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Trigonometry Examples
Step 1
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 2
Step 2.1
The exact value of is .
Step 3
Step 3.1
Add to both sides of the equation.
Step 3.2
To write as a fraction with a common denominator, multiply by .
Step 3.3
To write as a fraction with a common denominator, multiply by .
Step 3.4
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Step 3.4.1
Multiply by .
Step 3.4.2
Multiply by .
Step 3.4.3
Multiply by .
Step 3.4.4
Multiply by .
Step 3.5
Combine the numerators over the common denominator.
Step 3.6
Simplify the numerator.
Step 3.6.1
Move to the left of .
Step 3.6.2
Move to the left of .
Step 3.6.3
Add and .
Step 4
Step 4.1
Divide each term in by .
Step 4.2
Simplify the left side.
Step 4.2.1
Cancel the common factor of .
Step 4.2.1.1
Cancel the common factor.
Step 4.2.1.2
Divide by .
Step 4.3
Simplify the right side.
Step 4.3.1
Multiply the numerator by the reciprocal of the denominator.
Step 4.3.2
Multiply .
Step 4.3.2.1
Multiply by .
Step 4.3.2.2
Multiply by .
Step 5
The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from to find the solution in the fourth quadrant.
Step 6
Step 6.1
Simplify .
Step 6.1.1
To write as a fraction with a common denominator, multiply by .
Step 6.1.2
Combine fractions.
Step 6.1.2.1
Combine and .
Step 6.1.2.2
Combine the numerators over the common denominator.
Step 6.1.3
Simplify the numerator.
Step 6.1.3.1
Multiply by .
Step 6.1.3.2
Subtract from .
Step 6.2
Move all terms not containing to the right side of the equation.
Step 6.2.1
Add to both sides of the equation.
Step 6.2.2
To write as a fraction with a common denominator, multiply by .
Step 6.2.3
To write as a fraction with a common denominator, multiply by .
Step 6.2.4
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Step 6.2.4.1
Multiply by .
Step 6.2.4.2
Multiply by .
Step 6.2.4.3
Multiply by .
Step 6.2.4.4
Multiply by .
Step 6.2.5
Combine the numerators over the common denominator.
Step 6.2.6
Simplify the numerator.
Step 6.2.6.1
Multiply by .
Step 6.2.6.2
Move to the left of .
Step 6.2.6.3
Add and .
Step 6.3
Divide each term in by and simplify.
Step 6.3.1
Divide each term in by .
Step 6.3.2
Simplify the left side.
Step 6.3.2.1
Cancel the common factor of .
Step 6.3.2.1.1
Cancel the common factor.
Step 6.3.2.1.2
Divide by .
Step 6.3.3
Simplify the right side.
Step 6.3.3.1
Multiply the numerator by the reciprocal of the denominator.
Step 6.3.3.2
Multiply .
Step 6.3.3.2.1
Multiply by .
Step 6.3.3.2.2
Multiply by .
Step 7
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 8
The period of the function is so values will repeat every radians in both directions.
, for any integer