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Trigonometry Examples
Step 1
Step 1.1
Simplify each term.
Step 1.1.1
Rewrite in terms of sines and cosines, then cancel the common factors.
Step 1.1.1.1
Add parentheses.
Step 1.1.1.2
Reorder and .
Step 1.1.1.3
Rewrite in terms of sines and cosines.
Step 1.1.1.4
Cancel the common factors.
Step 1.1.2
Rewrite in terms of sines and cosines.
Step 1.1.3
Combine and .
Step 2
Multiply both sides of the equation by .
Step 3
Apply the distributive property.
Step 4
Rewrite using the commutative property of multiplication.
Step 5
Step 5.1
Cancel the common factor.
Step 5.2
Rewrite the expression.
Step 6
Step 6.1
Reorder and .
Step 6.2
Reorder and .
Step 6.3
Apply the sine double-angle identity.
Step 7
Multiply by .
Step 8
Apply the sine double-angle identity.
Step 9
Step 9.1
Factor out of .
Step 9.2
Factor out of .
Step 9.3
Factor out of .
Step 10
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 11
Step 11.1
Set equal to .
Step 11.2
Solve for .
Step 11.2.1
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 11.2.2
Simplify the right side.
Step 11.2.2.1
The exact value of is .
Step 11.2.3
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 11.2.4
Subtract from .
Step 11.2.5
Find the period of .
Step 11.2.5.1
The period of the function can be calculated using .
Step 11.2.5.2
Replace with in the formula for period.
Step 11.2.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 11.2.5.4
Divide by .
Step 11.2.6
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 12
Step 12.1
Set equal to .
Step 12.2
Solve for .
Step 12.2.1
Subtract from both sides of the equation.
Step 12.2.2
Divide each term in by and simplify.
Step 12.2.2.1
Divide each term in by .
Step 12.2.2.2
Simplify the left side.
Step 12.2.2.2.1
Cancel the common factor of .
Step 12.2.2.2.1.1
Cancel the common factor.
Step 12.2.2.2.1.2
Divide by .
Step 12.2.2.3
Simplify the right side.
Step 12.2.2.3.1
Move the negative in front of the fraction.
Step 12.2.3
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 12.2.4
Simplify the right side.
Step 12.2.4.1
The exact value of is .
Step 12.2.5
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 12.2.6
Simplify .
Step 12.2.6.1
To write as a fraction with a common denominator, multiply by .
Step 12.2.6.2
Combine fractions.
Step 12.2.6.2.1
Combine and .
Step 12.2.6.2.2
Combine the numerators over the common denominator.
Step 12.2.6.3
Simplify the numerator.
Step 12.2.6.3.1
Multiply by .
Step 12.2.6.3.2
Subtract from .
Step 12.2.7
Find the period of .
Step 12.2.7.1
The period of the function can be calculated using .
Step 12.2.7.2
Replace with in the formula for period.
Step 12.2.7.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 12.2.7.4
Divide by .
Step 12.2.8
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 13
The final solution is all the values that make true.
, for any integer
Step 14
Consolidate and to .
, for any integer