Trigonometry Examples

Solve for t 6=-8cos(pi/6t)
Step 1
Rewrite the equation as .
Step 2
Divide each term in by and simplify.
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Step 2.1
Divide each term in by .
Step 2.2
Simplify the left side.
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Step 2.2.1
Cancel the common factor of .
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Step 2.2.1.1
Cancel the common factor.
Step 2.2.1.2
Divide by .
Step 2.3
Simplify the right side.
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Step 2.3.1
Cancel the common factor of and .
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Step 2.3.1.1
Factor out of .
Step 2.3.1.2
Cancel the common factors.
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Step 2.3.1.2.1
Factor out of .
Step 2.3.1.2.2
Cancel the common factor.
Step 2.3.1.2.3
Rewrite the expression.
Step 2.3.2
Move the negative in front of the fraction.
Step 3
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 4
Simplify the left side.
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Step 4.1
Combine and .
Step 5
Simplify the right side.
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Step 5.1
Evaluate .
Step 6
Multiply both sides of the equation by .
Step 7
Simplify both sides of the equation.
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Step 7.1
Simplify the left side.
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Step 7.1.1
Simplify .
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Step 7.1.1.1
Cancel the common factor of .
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Step 7.1.1.1.1
Cancel the common factor.
Step 7.1.1.1.2
Rewrite the expression.
Step 7.1.1.2
Cancel the common factor of .
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Step 7.1.1.2.1
Factor out of .
Step 7.1.1.2.2
Cancel the common factor.
Step 7.1.1.2.3
Rewrite the expression.
Step 7.2
Simplify the right side.
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Step 7.2.1
Simplify .
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Step 7.2.1.1
Multiply .
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Step 7.2.1.1.1
Combine and .
Step 7.2.1.1.2
Multiply by .
Step 7.2.1.2
Replace with an approximation.
Step 7.2.1.3
Divide by .
Step 8
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 9
Solve for .
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Step 9.1
Multiply both sides of the equation by .
Step 9.2
Simplify both sides of the equation.
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Step 9.2.1
Simplify the left side.
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Step 9.2.1.1
Simplify .
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Step 9.2.1.1.1
Cancel the common factor of .
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Step 9.2.1.1.1.1
Cancel the common factor.
Step 9.2.1.1.1.2
Rewrite the expression.
Step 9.2.1.1.2
Cancel the common factor of .
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Step 9.2.1.1.2.1
Factor out of .
Step 9.2.1.1.2.2
Cancel the common factor.
Step 9.2.1.1.2.3
Rewrite the expression.
Step 9.2.2
Simplify the right side.
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Step 9.2.2.1
Simplify .
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Step 9.2.2.1.1
Multiply by .
Step 9.2.2.1.2
Subtract from .
Step 9.2.2.1.3
Multiply .
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Step 9.2.2.1.3.1
Combine and .
Step 9.2.2.1.3.2
Multiply by .
Step 9.2.2.1.4
Replace with an approximation.
Step 9.2.2.1.5
Divide by .
Step 10
Find the period of .
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Step 10.1
The period of the function can be calculated using .
Step 10.2
Replace with in the formula for period.
Step 10.3
is approximately which is positive so remove the absolute value
Step 10.4
Multiply the numerator by the reciprocal of the denominator.
Step 10.5
Cancel the common factor of .
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Step 10.5.1
Factor out of .
Step 10.5.2
Cancel the common factor.
Step 10.5.3
Rewrite the expression.
Step 10.6
Multiply by .
Step 11
The period of the function is so values will repeat every radians in both directions.
, for any integer