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Trigonometry Examples
Step 1
Step 1.1
Evaluate .
Step 1.2
Move to the left of .
Step 1.3
Evaluate .
Step 1.4
Multiply by .
Step 2
Use the identity to solve the equation. In this identity, represents the angle created by plotting point on a graph and therefore can be found using .
where and
Step 3
Set up the equation to find the value of .
Step 4
Step 4.1
Divide by .
Step 4.2
Multiply by .
Step 4.3
Evaluate .
Step 5
Step 5.1
Raise to the power of .
Step 5.2
Raise to the power of .
Step 5.3
Add and .
Step 6
Substitute the known values into the equation.
Step 7
Step 7.1
Divide each term in by .
Step 7.2
Simplify the left side.
Step 7.2.1
Cancel the common factor of .
Step 7.2.1.1
Cancel the common factor.
Step 7.2.1.2
Divide by .
Step 7.3
Simplify the right side.
Step 7.3.1
Multiply the numerator by the reciprocal of the denominator.
Step 7.3.2
Multiply by .
Step 7.3.3
Combine and simplify the denominator.
Step 7.3.3.1
Multiply by .
Step 7.3.3.2
Raise to the power of .
Step 7.3.3.3
Raise to the power of .
Step 7.3.3.4
Use the power rule to combine exponents.
Step 7.3.3.5
Add and .
Step 7.3.3.6
Rewrite as .
Step 7.3.3.6.1
Use to rewrite as .
Step 7.3.3.6.2
Apply the power rule and multiply exponents, .
Step 7.3.3.6.3
Combine and .
Step 7.3.3.6.4
Cancel the common factor of .
Step 7.3.3.6.4.1
Cancel the common factor.
Step 7.3.3.6.4.2
Rewrite the expression.
Step 7.3.3.6.5
Evaluate the exponent.
Step 7.3.4
Evaluate the root.
Step 7.3.5
Divide by .
Step 7.3.6
Multiply by .
Step 8
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 9
Step 9.1
The exact value of is .
Step 10
Step 10.1
Add to both sides of the equation.
Step 10.2
To write as a fraction with a common denominator, multiply by .
Step 10.3
To write as a fraction with a common denominator, multiply by .
Step 10.4
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Step 10.4.1
Multiply by .
Step 10.4.2
Multiply by .
Step 10.4.3
Multiply by .
Step 10.4.4
Multiply by .
Step 10.5
Combine the numerators over the common denominator.
Step 10.6
Simplify the numerator.
Step 10.6.1
Move to the left of .
Step 10.6.2
Move to the left of .
Step 10.6.3
Add and .
Step 11
The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from to find the solution in the second quadrant.
Step 12
Step 12.1
Simplify .
Step 12.1.1
To write as a fraction with a common denominator, multiply by .
Step 12.1.2
Combine fractions.
Step 12.1.2.1
Combine and .
Step 12.1.2.2
Combine the numerators over the common denominator.
Step 12.1.3
Simplify the numerator.
Step 12.1.3.1
Move to the left of .
Step 12.1.3.2
Subtract from .
Step 12.2
Move all terms not containing to the right side of the equation.
Step 12.2.1
Add to both sides of the equation.
Step 12.2.2
To write as a fraction with a common denominator, multiply by .
Step 12.2.3
To write as a fraction with a common denominator, multiply by .
Step 12.2.4
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Step 12.2.4.1
Multiply by .
Step 12.2.4.2
Multiply by .
Step 12.2.4.3
Multiply by .
Step 12.2.4.4
Multiply by .
Step 12.2.5
Combine the numerators over the common denominator.
Step 12.2.6
Simplify the numerator.
Step 12.2.6.1
Multiply by .
Step 12.2.6.2
Move to the left of .
Step 12.2.6.3
Add and .
Step 13
Step 13.1
The period of the function can be calculated using .
Step 13.2
Replace with in the formula for period.
Step 13.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 13.4
Divide by .
Step 14
The period of the function is so values will repeat every radians in both directions.
, for any integer