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Trigonometry Examples
Step 1
Step 1.1
Subtract from both sides of the equation.
Step 1.2
Subtract from both sides of the equation.
Step 2
Step 2.1
Factor out of .
Step 2.2
Factor out of .
Step 2.3
Factor out of .
Step 2.4
Rearrange terms.
Step 2.5
Apply pythagorean identity.
Step 2.6
Simplify each term.
Step 2.6.1
Rewrite as .
Step 2.6.2
Expand using the FOIL Method.
Step 2.6.2.1
Apply the distributive property.
Step 2.6.2.2
Apply the distributive property.
Step 2.6.2.3
Apply the distributive property.
Step 2.6.3
Simplify and combine like terms.
Step 2.6.3.1
Simplify each term.
Step 2.6.3.1.1
Multiply .
Step 2.6.3.1.1.1
Raise to the power of .
Step 2.6.3.1.1.2
Raise to the power of .
Step 2.6.3.1.1.3
Use the power rule to combine exponents.
Step 2.6.3.1.1.4
Add and .
Step 2.6.3.1.2
Rewrite using the commutative property of multiplication.
Step 2.6.3.1.3
Multiply .
Step 2.6.3.1.3.1
Multiply by .
Step 2.6.3.1.3.2
Multiply by .
Step 2.6.3.1.3.3
Raise to the power of .
Step 2.6.3.1.3.4
Raise to the power of .
Step 2.6.3.1.3.5
Use the power rule to combine exponents.
Step 2.6.3.1.3.6
Add and .
Step 2.6.3.2
Reorder the factors of .
Step 2.6.3.3
Subtract from .
Step 2.6.4
Move .
Step 2.6.5
Rearrange terms.
Step 2.6.6
Apply pythagorean identity.
Step 2.6.7
Multiply by .
Step 2.7
Combine the opposite terms in .
Step 2.7.1
Subtract from .
Step 2.7.2
Add and .
Step 3
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 4
Step 4.1
Set equal to .
Step 4.2
Solve for .
Step 4.2.1
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 4.2.2
Simplify the right side.
Step 4.2.2.1
The exact value of is .
Step 4.2.3
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 4.2.4
Simplify .
Step 4.2.4.1
To write as a fraction with a common denominator, multiply by .
Step 4.2.4.2
Combine fractions.
Step 4.2.4.2.1
Combine and .
Step 4.2.4.2.2
Combine the numerators over the common denominator.
Step 4.2.4.3
Simplify the numerator.
Step 4.2.4.3.1
Multiply by .
Step 4.2.4.3.2
Subtract from .
Step 4.2.5
Find the period of .
Step 4.2.5.1
The period of the function can be calculated using .
Step 4.2.5.2
Replace with in the formula for period.
Step 4.2.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 4.2.5.4
Divide by .
Step 4.2.6
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 5
Step 5.1
Set equal to .
Step 5.2
Solve for .
Step 5.2.1
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 5.2.2
Simplify the right side.
Step 5.2.2.1
The exact value of is .
Step 5.2.3
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 5.2.4
Subtract from .
Step 5.2.5
Find the period of .
Step 5.2.5.1
The period of the function can be calculated using .
Step 5.2.5.2
Replace with in the formula for period.
Step 5.2.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 5.2.5.4
Divide by .
Step 5.2.6
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 6
The final solution is all the values that make true.
, for any integer
Step 7
Consolidate the answers.
, for any integer