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Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742

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BuyFindarrow_forward

Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742
Textbook Problem

SKILLS

9-14 ■ Finding the Equation for a Rotated Conic Determine the equation of the given conic in XY-coordinates when the coordinate axes are rotated through the indicated angle.

x 2 + 2 y 2 = 16 , ϕ = sin 1 3 5

To determine

The equation of the conic x2+2y2=16 in XY-coordinates when the coordinate axes are rotated at the angle ϕ=sin135.

Explanation

Given:

The equation of the conic x2+2y2=16 in XY-coordinates when the coordinate axes are rotated at the angle ϕ=sin135.

Approach:

The x-axes and y-axes in a coordinate plane are rotated through the acute angle ϕ to produce the X-axis and Y-axis. Then, the coordinates (x,y) and (X,Y) of a point in the xy and the XY-planes are

x=XcosϕYsinϕ …… (1)

y=Xsinϕ+Ycosϕ …… (2)

Here, (x,y) is the coordinates of the point and ϕ is the angle at which coordinate axis are rotated.

Write the formula of the inverse trigonometry.

sin2θ=1cos2θ

sin(sin1x)=x

Calculation:

Substitute sin135 for ϕ in equation (1).

x=Xcos(sin135)Ysin(sin135)=X(1sin2(sin135))Ysin(sin135)=X(1{sin(sin135)}2)Ysin(sin135)=X(1(35)2)Y(35)

Further solve,

x=X(1925)Y(35)=1625X35Y=45X35Y

Similarly, Substitute sin135 for ϕ in equation (2).

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