Calculus Examples

Find the Horizontal Tangent Line y=sin(x)
Step 1
Set as a function of .
Step 2
The derivative of with respect to is .
Step 3
Set the derivative equal to then solve the equation .
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Step 3.1
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 3.2
Simplify the right side.
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Step 3.2.1
The exact value of is .
Step 3.3
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 3.4
Simplify .
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Step 3.4.1
To write as a fraction with a common denominator, multiply by .
Step 3.4.2
Combine fractions.
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Step 3.4.2.1
Combine and .
Step 3.4.2.2
Combine the numerators over the common denominator.
Step 3.4.3
Simplify the numerator.
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Step 3.4.3.1
Multiply by .
Step 3.4.3.2
Subtract from .
Step 3.5
Find the period of .
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Step 3.5.1
The period of the function can be calculated using .
Step 3.5.2
Replace with in the formula for period.
Step 3.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 3.5.4
Divide by .
Step 3.6
The period of the function is so values will repeat every radians in both directions.
, for any integer
Step 3.7
Consolidate the answers.
, for any integer
, for any integer
Step 4
Solve the original function at .
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Step 4.1
Replace the variable with in the expression.
Step 4.2
Simplify the result.
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Step 4.2.1
The exact value of is .
Step 4.2.2
The final answer is .
Step 5
Solve the original function at .
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Step 5.1
Replace the variable with in the expression.
Step 5.2
Simplify the result.
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Step 5.2.1
To write as a fraction with a common denominator, multiply by .
Step 5.2.2
Combine fractions.
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Step 5.2.2.1
Combine and .
Step 5.2.2.2
Combine the numerators over the common denominator.
Step 5.2.3
Simplify the numerator.
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Step 5.2.3.1
Move to the left of .
Step 5.2.3.2
Add and .
Step 5.2.4
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the fourth quadrant.
Step 5.2.5
The exact value of is .
Step 5.2.6
Multiply by .
Step 5.2.7
The final answer is .
Step 6
The horizontal tangent line on function is .
Step 7