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Trigonometry Examples
Step 1
Step 1.1
The exact value of is .
Step 1.1.1
Rewrite as an angle where the values of the six trigonometric functions are known divided by .
Step 1.1.2
Apply the cosine half-angle identity .
Step 1.1.3
Change the to because cosine is positive in the first quadrant.
Step 1.1.4
Simplify .
Step 1.1.4.1
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.
Step 1.1.4.2
The exact value of is .
Step 1.1.4.3
Write as a fraction with a common denominator.
Step 1.1.4.4
Combine the numerators over the common denominator.
Step 1.1.4.5
Multiply the numerator by the reciprocal of the denominator.
Step 1.1.4.6
Multiply .
Step 1.1.4.6.1
Multiply by .
Step 1.1.4.6.2
Multiply by .
Step 1.1.4.7
Rewrite as .
Step 1.1.4.8
Simplify the denominator.
Step 1.1.4.8.1
Rewrite as .
Step 1.1.4.8.2
Pull terms out from under the radical, assuming positive real numbers.
Step 1.2
Write as a fraction with a common denominator.
Step 1.3
Combine the numerators over the common denominator.
Step 1.4
Multiply the numerator by the reciprocal of the denominator.
Step 1.5
Multiply .
Step 1.5.1
Multiply by .
Step 1.5.2
Multiply by .
Step 1.6
Rewrite as .
Step 1.7
Simplify the denominator.
Step 1.7.1
Rewrite as .
Step 1.7.2
Pull terms out from under the radical, assuming positive real numbers.
Step 2
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 3
Step 3.1
Evaluate .
Step 4
Multiply both sides of the equation by .
Step 5
Step 5.1
Simplify the left side.
Step 5.1.1
Cancel the common factor of .
Step 5.1.1.1
Cancel the common factor.
Step 5.1.1.2
Rewrite the expression.
Step 5.2
Simplify the right side.
Step 5.2.1
Multiply by .
Step 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 7
Step 7.1
Multiply both sides of the equation by .
Step 7.2
Simplify both sides of the equation.
Step 7.2.1
Simplify the left side.
Step 7.2.1.1
Cancel the common factor of .
Step 7.2.1.1.1
Cancel the common factor.
Step 7.2.1.1.2
Rewrite the expression.
Step 7.2.2
Simplify the right side.
Step 7.2.2.1
Simplify .
Step 7.2.2.1.1
Subtract from .
Step 7.2.2.1.2
Multiply by .
Step 8
Step 8.1
The period of the function can be calculated using .
Step 8.2
Replace with in the formula for period.
Step 8.3
is approximately which is positive so remove the absolute value
Step 8.4
Multiply the numerator by the reciprocal of the denominator.
Step 8.5
Multiply by .
Step 9
The period of the function is so values will repeat every degrees in both directions.
, for any integer