Trigonometry Examples

Solve for θ in Radians sin(3theta)=-1
Step 1
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 2
Simplify the right side.
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Step 2.1
The exact value of is .
Step 3
Divide each term in by and simplify.
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Step 3.1
Divide each term in by .
Step 3.2
Simplify the left side.
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Step 3.2.1
Cancel the common factor of .
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Step 3.2.1.1
Cancel the common factor.
Step 3.2.1.2
Divide by .
Step 3.3
Simplify the right side.
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Step 3.3.1
Multiply the numerator by the reciprocal of the denominator.
Step 3.3.2
Multiply .
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Step 3.3.2.1
Multiply by .
Step 3.3.2.2
Multiply by .
Step 4
The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from , to find a reference angle. Next, add this reference angle to to find the solution in the third quadrant.
Step 5
Simplify the expression to find the second solution.
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Step 5.1
Subtract from .
Step 5.2
The resulting angle of is positive, less than , and coterminal with .
Step 5.3
Divide each term in by and simplify.
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Step 5.3.1
Divide each term in by .
Step 5.3.2
Simplify the left side.
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Step 5.3.2.1
Cancel the common factor of .
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Step 5.3.2.1.1
Cancel the common factor.
Step 5.3.2.1.2
Divide by .
Step 5.3.3
Simplify the right side.
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Step 5.3.3.1
Multiply the numerator by the reciprocal of the denominator.
Step 5.3.3.2
Cancel the common factor of .
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Step 5.3.3.2.1
Factor out of .
Step 5.3.3.2.2
Cancel the common factor.
Step 5.3.3.2.3
Rewrite the expression.
Step 6
Find the period of .
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Step 6.1
The period of the function can be calculated using .
Step 6.2
Replace with in the formula for period.
Step 6.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 7
Add to every negative angle to get positive angles.
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Step 7.1
Add to to find the positive angle.
Step 7.2
To write as a fraction with a common denominator, multiply by .
Step 7.3
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 7.3.1
Multiply by .
Step 7.3.2
Multiply by .
Step 7.4
Combine the numerators over the common denominator.
Step 7.5
Simplify the numerator.
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Step 7.5.1
Multiply by .
Step 7.5.2
Subtract from .
Step 7.6
Cancel the common factor of and .
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Step 7.6.1
Factor out of .
Step 7.6.2
Cancel the common factors.
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Step 7.6.2.1
Factor out of .
Step 7.6.2.2
Cancel the common factor.
Step 7.6.2.3
Rewrite the expression.
Step 7.7
List the new angles.
Step 8
The period of the function is so values will repeat every radians in both directions.
, for any integer
Step 9
Consolidate the answers.
, for any integer