Find Slope of the tangent #calculus #maths #anglebetween #differentiation #mathematics

🌟 Understanding the Angle Between the Radius Vector and Tangent for Polar Curves! 🌟

In this insightful video, we dive deep into the fascinating world of polar coordinates as we explore the curve defined by the equation r=a(1+cosθ). We’ll take you step-by-step through the process of finding the angle Φ between the radius vector and the tangent line at a specific point—specifically at .

🔍 What You’ll Learn:

How to differentiate polar curves using logarithmic identities
The relationship between the radius vector, tangent line, and their angles
Step-by-step calculations leading to the angle Φ
A clear understanding of the slope of the tangent line

Angle blw radius vector and tangent & also find slope of the tangent as indicated for curve
r = a(1 + cosθ) at θ = π/3.

take log on b.s
log r = log a(1 + cosθ)
log r = log a + log(1 + cosθ)
log a*b = log a + log b
differentiate w.r.t θ
1/r * dr/dθ = 0 + 1/(1 + cosθ) * (0 + (-sinθ))
cot Φ = -sinθ/(1 + cosθ)
-sinθ = 2 sin(θ/2) cos(θ/2)
1 + cosθ = 2 cos²(θ/2)
cot Φ = -2 sin(θ/2) cos(θ/2) / 2 cos²(θ/2)
cot Φ = -2 sin(θ/2) cos(θ/2) / 2 cos(θ/2) cos(θ/2)
cot Φ = -sin(θ/2) / cos(θ/2)
cot Φ = -tan(θ/2)
-tan(θ/2) = cot(π/2 + θ/2)
cot Φ = cot(π/2 + θ/2)
Φ = π/2 + θ/2
at θ = π/3
Φ = π/2 + π/6
Φ = 2π/3

Whether you're a math enthusiast, a student looking for extra help, or someone curious about polar curves, this video is packed with valuable insights and practical examples. Join us as we break down complex concepts into easily digestible parts!

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