Get the answer to "What is the equation of the tangent line..."

High School
Calculus and Analysis
Calculus 1
Derivatives

Jazmin Strong

Answered question

2022-04-07

What is the equation of the tangent line of

f (x) = \cos x \sin x

x = \frac{π}{3}

Answer & Explanation

o2z1zsjj0

Beginner2022-04-08Added 10 answers

f (\frac{π}{3}) = \cos (\frac{π}{3}) \sin (\frac{π}{3}) = (\frac{1}{2}) (\frac{\sqrt{3}}{2}) = \frac{\sqrt{3}}{4}

.
So the line we seek contains the point

(\frac{π}{3}, \frac{\sqrt{3}}{4})

.
The slope is given by the derivative. We could differentiate by using the chain rule, but let's use some trigonometry to rewrite the function instead.

\sin (2 θ) = 2 \sin θ \cos θ

,
so

f (x) = \cos x \sin x = \frac{1}{2} \sin (2 x)

And

f^{'} (x) = \cos (2 x)

x = \frac{π}{3}

we get slope

= \cos (\frac{2 π}{3}) = - \frac{1}{2}

The line through

(\frac{π}{3}, \frac{\sqrt{3}}{4})

, with slope

m = - \frac{1}{2}

y = - \frac{1}{2} x + \frac{π}{6} + \frac{\sqrt{3}}{4}

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