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Trigonometry Examples
Step 1
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 2
Step 2.1
Set equal to .
Step 2.2
Solve for .
Step 2.2.1
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 2.2.2
Simplify .
Step 2.2.2.1
Rewrite as .
Step 2.2.2.2
Pull terms out from under the radical, assuming positive real numbers.
Step 2.2.2.3
Plus or minus is .
Step 2.2.3
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 2.2.4
Simplify the right side.
Step 2.2.4.1
The exact value of is .
Step 2.2.5
Add to both sides of the equation.
Step 2.2.6
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 2.2.7
Solve for .
Step 2.2.7.1
Simplify .
Step 2.2.7.1.1
To write as a fraction with a common denominator, multiply by .
Step 2.2.7.1.2
Combine fractions.
Step 2.2.7.1.2.1
Combine and .
Step 2.2.7.1.2.2
Combine the numerators over the common denominator.
Step 2.2.7.1.3
Simplify the numerator.
Step 2.2.7.1.3.1
Multiply by .
Step 2.2.7.1.3.2
Subtract from .
Step 2.2.7.2
Add to both sides of the equation.
Step 2.2.8
Find the period of .
Step 2.2.8.1
The period of the function can be calculated using .
Step 2.2.8.2
Replace with in the formula for period.
Step 2.2.8.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 2.2.8.4
Divide by .
Step 2.2.9
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 3
Step 3.1
Set equal to .
Step 3.2
Solve for .
Step 3.2.1
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 3.2.2
Simplify the right side.
Step 3.2.2.1
The exact value of is .
Step 3.2.3
Add to both sides of the equation.
Step 3.2.4
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 3.2.5
Solve for .
Step 3.2.5.1
Simplify .
Step 3.2.5.1.1
To write as a fraction with a common denominator, multiply by .
Step 3.2.5.1.2
Combine fractions.
Step 3.2.5.1.2.1
Combine and .
Step 3.2.5.1.2.2
Combine the numerators over the common denominator.
Step 3.2.5.1.3
Simplify the numerator.
Step 3.2.5.1.3.1
Multiply by .
Step 3.2.5.1.3.2
Subtract from .
Step 3.2.5.2
Add to both sides of the equation.
Step 3.2.6
Find the period of .
Step 3.2.6.1
The period of the function can be calculated using .
Step 3.2.6.2
Replace with in the formula for period.
Step 3.2.6.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 3.2.6.4
Divide by .
Step 3.2.7
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 4
The final solution is all the values that make true.
, for any integer
Step 5
Consolidate and to .
, for any integer