How do you find the equation of the line tangent to #y=sinx# at #(pi/4, sqrt(2)/2)#?

Answer 1

#y=(4x-pi+4)/(4sqrt2)#

The slope of the tangent line can be found using the derivative of the function:

#dy/dx=cosx#
The slope of the tangent line at #x=pi/4# can then be found through plugging #pi/4# into the derivative.
#cos(pi/4)=sqrt2/2#
So, we have a line that passes through the point #(pi/4,sqrt2/2)# and a slope of #sqrt2/2#:
#y-y_0=m(x-x_0)#
#y-sqrt2/2=sqrt2/2(x-pi/4)#
#y=sqrt2/2x-(pisqrt2)/8+sqrt2/2#
#y=(4sqrt2x-pisqrt2+4sqrt2)/8#
Note that #sqrt2/8=sqrt2/(4sqrt2sqrt2)=1/(4sqrt2)#:
#y=(4x-pi+4)/(4sqrt2)#
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Answer 2

To find the equation of the line tangent to the curve y = sin(x) at the point (π/4, √2/2), we need to determine the slope of the tangent line at that point.

The slope of the tangent line can be found by taking the derivative of the function y = sin(x) with respect to x.

The derivative of sin(x) is cos(x).

Evaluating cos(x) at x = π/4, we get cos(π/4) = √2/2.

Therefore, the slope of the tangent line at (π/4, √2/2) is √2/2.

Using the point-slope form of a linear equation, we can write the equation of the tangent line as y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope.

Plugging in the values, we have y - √2/2 = (√2/2)(x - π/4).

Simplifying, the equation of the tangent line is y = (√2/2)x - (√2/2)(π/4) + √2/2.

This can be further simplified to y = (√2/2)x - (√2π/8) + √2/2.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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