Answer:
Step-by-step explanation:
Hi there,
To get started, recall that the cosθ of an angle has a definite maximum value, which is:
[tex]x_m_a_x = cos\theta\leq 1[/tex] Hence the cosine of any angle can never be greater than 1, which is illustrated if you plot the graph of cos(x) on a graphing calculator or Desmos.com. Thus, 1 is the maximum of cosθ; the same goes for sinθ, but at a different angle.
With this in mind, plug this in:
[tex]r = 3 + (3)cos\theta=3+(3)(1) = 3+3 =6\\r=6[/tex]
Therefore, the maximum r-value from this polar equation is 6.
If you are adept in calculus, this equation can also be solved using a derivative and some trigonometry:
[tex]\frac{d}{d\theta} [r]=\frac{dr}{d\theta} = \frac{d}{d\theta} (3)+(3)\frac{d}{d\theta} (cos\theta)\\\frac{dr}{d\theta} =0+(3)(-sin\theta)\\[/tex]
In calculus, the maximum of a function is when its rate is equal to zero:
[tex]\\\frac{dr}{d\theta}=0=0+(3)(-sin\theta)\\0=-3sin\theta\\0=sin\theta[/tex]
Sine of a RADIANS angle is only equal to zero when θ is equal to the following:
[tex]\theta=\pi *n[/tex] where n=0,1,2,3,... and π is in radians
Hence plug in 0 into the original equation for θ to make cosine a maximum:
r=3 + 3(cos0) = 3 +3(1) = 6
Same answer obtained!
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