Answer:
[tex]x^2+y^2=16[/tex]
Step-by-step explanation:
Given: [tex]x=4\cos t, y=4\sin t[/tex]
We need to eliminate the parameter t from parametric equation.
[tex]x=4\cos t[/tex]
Square both sides
[tex]x^2=(4\cos t)^2[/tex]
[tex]x^2=16\cos^2 t[/tex]
[tex]y=4\sin t[/tex]
Square both sides
[tex]y^2=(4\sin t)^2[/tex]
[tex]y^2=16\sin^2 t[/tex]
Add both equation
[tex]x^2+y^2=16\cos^2t+16\sin^2t[/tex]
[tex]x^2+y^2=16(\cos^2t+\sin^2t)[/tex] [tex]\because \sin^2\theta+\cos^2\theta = 1[/tex]
[tex]x^2+y^2=16[/tex]
Hence, The new equation without parameter t would be [tex]x^2+y^2=16[/tex]
After eliminating the parameter the equation can be written as x²+y² =16.
[tex]Sin \theta=\dfrac{Perpendicular}{Hypotenuse}[/tex]
[tex]Cos \theta=\dfrac{Base}{Hypotenuse}[/tex]
[tex]Tan \theta=\dfrac{Perpendicular}{Base}[/tex]
where perpendicular is the side of the triangle which is opposite to the angle, and the hypotenuse is the longest side of the triangle which is opposite to the 90° angle.
We need to eliminate the parameter t from equations, therefore, we can write for x,
Given to us
x = 4 cos t
y = 4 sin t
[tex]x = 4 cos t\\\\\text{Squaring both the sides}\\\\x^2 = 16\cos^2t[/tex]
Rewriting the equation of y,
[tex]y = 4 sin t\\\\\text{Squaring both the sides}\\\\y^2 = 16\ sin^2t[/tex]
Sum of both the equations
When we Add both the equations we get,
[tex]x^2+y^2 = 16 cos^2 t+16 sin^2t\\\\\text{Taking 16 as the common factor}\\\\x^2+y^2 = 16 (cos^2 t+sin^2t)\\\\[/tex]
Using the trigonometric identity,
[tex]cos^2 x+sin^2x=1[/tex]
Therefore,
[tex]x^2+y^2 = 16 (cos^2 t+sin^2t)\\\\x^2+y^2 = 16 (1)\\\\x^2+y^2 = 16[/tex]
Hence, After eliminating the parameter the equation can be written as x²+y² =16.
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