Mathematics: The curve given by x = sin(t) and y = sin(t + sin(t)) has two tangent lines at the point (x,y) = (0,0). List both of them in order of increasing slope. Your answers should be in the form of y = f(x) without t s. Line with smaller slope: y(x): Line with larger slope: y(x):

Box 6 (rightmost) into slot 1 Box 5 into slot 2 Box 4 into slot 3 Box 2 into slot 4...

The curve given by x = sin(t) and y = sin(t + sin(t))

MzThixkumz

01.02.2022

slope of the tangent

$\frac{d y}{d x} = \frac{cos(x+sin^{-1}(x) (1+\sqrt{1-x^{2}) } }{\sqrt{1-x^{2} } }$

The function y = f(x)

y(x) = 2 x + C

Step-by-step explanation:

Step(i):-

Given x= sin t ...(i)

Differentiating equation (i) with respective to 'x' , we get

$\frac{d x}{d t} = cost$

Given y = sin ( t + sin (t)) ...(ii)

Differentiating equation (ii) with respective to 'x' , we get

$\frac{d y}{d t} = cos (t + sin t ) (1 + cos t)$

Step(ii):-

$\frac{dy}{dx} = \frac{\frac{dy}{dt} }{\frac{dx}{dt} }$

$\frac{d y}{d x} = \frac{cos(t+sin t) (1+cost)}{cost}$

we know that

x = sin t

t = sin⁻¹ (x)

cost = √1 - sin²(t) = (√1-x²)

$\frac{d y}{d x} = \frac{cos(x+sin^{-1}(x) (1+\sqrt{1-x^{2}) } }{\sqrt{1-x^{2} } }$

Put x =0 and y=0

$\frac{d y}{d x} = 2$

d y = 2 d x

Integrating with respective to 'x' , we get

y(x) = 2 x + C

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