How do you find an equation of the tangent line to the curve at the given point: #y = csc (x) - 2 sin (x)# and P = (pi/6, 1)?

Answer 1

The equation is #y = -3sqrt(3)x + (pisqrt(3) + 1)/2#

Start by differentiating the function.

#y= 1/sinx - 2sinx#
#dy/dx= (0 xx sinx - cosx xx 1)/(sinx)^2 - 2cosx#
#dy/dx= -cosx/sin^2x - 2cosx#
#dy/dx = -cotxcscx - 2cosx#

Now, find the slope of the tangent.

#dy/dx = -cot(pi/6)csc(pi/6) - 2cos(pi/6)#
#dy/dx = -1/tan(pi/6) xx 1/sin(pi/6) - 2cos(pi/6)#
#dy/dx = -1/(1/sqrt(3)) xx 1/(1/2) - 2(sqrt(3)/2)#
#dy/dx= -sqrt(3) xx 2 - sqrt(3)#
#dy/dx = -2sqrt(3) - sqrt(3)#
#dy/dx= -3sqrt(3)#

We now find the equation of the tangent line with our point and our slope.

#y_ y_1 = m(x- x_1)#
#y - 1 = -3sqrt(3)(x - pi/6)#
#y - 1 = -3sqrt(3)x + (pisqrt(3))/2#
#y = -3sqrt(3)x + (pisqrt(3))/2 + 1#
#y = -3sqrt(3)x + (pisqrt(3) + 1)/2#

Hopefully this helps!

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Answer 2

To find the equation of the tangent line to the curve at the given point (pi/6, 1), we need to find the derivative of the function y = csc(x) - 2sin(x) and evaluate it at x = pi/6.

The derivative of y = csc(x) - 2sin(x) can be found using the chain rule and product rule.

The derivative is given by: dy/dx = -csc(x)cot(x) - 2cos(x)

Evaluating the derivative at x = pi/6, we have: dy/dx = -csc(pi/6)cot(pi/6) - 2cos(pi/6)

Simplifying further, we get: dy/dx = -2 - 2√3

Now, we have the slope of the tangent line at the point (pi/6, 1).

Using the point-slope form of a line, the equation of the tangent line is: y - 1 = (-2 - 2√3)(x - pi/6)

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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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