**Problem: Polar Curves and Tangents****Task:** Sketch the polar curve $ r^2 = \cos(6\theta) $ and compute the tangent line at the value $ \theta = \frac{\pi}{24} $.**Instructions:**1. Begin by understanding the polar equation $ r^2 = \cos(6\theta) $. - This equation expresses a relationship between the radius $ r $ and the angle $ \theta $ in polar coordinates. - It involves a cosine term with a multiple angle, which suggests the curve will have multiple symmetries or petals.2. To sketch the curve: - Evaluate $ r $ for several values of $ \theta $ ranging from 0 to $ 2\pi $. - Note when $ \cos(6\theta) $ results in positive, zero, and negative values to determine where the curve crosses the pole (origin) and where it is undefined.3. To find the tangent line at $ \theta = \frac{\pi}{24} $: - Calculate the derivative of $ r $ with respect to $ \theta $ to find the slope of the tangent. - Use the relationship $ \frac{dr}{d\theta} $ and evaluate it at $ \theta = \frac{\pi}{24} $.4. Consider converting to Cartesian coordinates if necessary for finding the equation of the tangent line: - Use the transformation $ x = r\cos(\theta) $ and $ y = r\sin(\theta) $.5. Analyze graphically: - Diagrams may illustrate how the curve behaves and where the tangent line is located for better visualization.By understanding these steps, you can accurately sketch and analyze the polar curve and compute the required tangent line.

Answered: Image /qna-images/answer/8c49ab5c-a9da-477d-bfa2-fdc489fbb6b5.jpg

Answered: Sketch me polar urve =cos(60) %3D and…

Math

Calculus

Sketch me polar urve =cos(60) %3D and wmpute the tangent linm ar thm Value = 24 TT

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

See similar textbooks

Related questions

Q: (b) cosec 0 sec 0 cot A = 1+tan²0

Q: length of the curve r = 4 sin 20 for ÷sos 6 2

A: To find the length of the curve r=4 sin 2θ for π6≤θ≤π2.

Q: Given that a = arcsin (-), sec B = and tan B < 0, find the exact value of sin(a +B).

A: Given that α=arcsin-23⇒sinα=-23secβ=54⇒1cosβ=54⇒cosβ=45,tanβ<0

Q: Find the length of the curve r = cos(f) + sin(0), 0sOs

A: The objective is to find the length of the curve.

Q: x= a cos4 theta y= a sin4 theta by the given equation, find the equation of a and b when theta is…

Q: 11.) cos os (sin-1 -tan-1%) 12.) tan (arccos %- arcsin (-12/13))

Q: cos x = 8/17, 3pi/2<x<2pi. Find cot x

A: cosx=817=BHthrefore by using right angle triangle and Pythagoras theoremH2=B2+P2172=82+P2P=225P=15

Q: 21 If cot 0 = find csc 0 and sin 0. 20

A: Consider a right triangle △ABC as shown in the figure, the cot θ is defined as, cotθ=ABACABAC=2120…

Q: Find 0, given sin'0 - 0.2 sin0 -0.01. (Give your answers correct to 2 decimal place).

Q: (5) Let v = 2 sine. a) Find where the curve horizontal tangent lines. b) Find where the vertical…

Q: Can you please give me two points from the topologist's sine curve?

A: Given: The topologist sine curve is given by, S=x, sin1x | 0<x≤1∪(0, 0) To find: Two points from…

Q: Given that sin = 0.9898, determine to four decimal places without a calculator, cos z

Q: 12. In a right triangle, the sine of an angle is equal to the cosine of its complement. O True False

A: Please see the explanation below

Q: Question 2 A is a fixed point on a circle centre O, radius 1 cm. P is variable point which moves…

Q: SKet the cerve r²= 29° cOS 20

Q: Find a decimal approximation to 100 sin (65) correct to 38 significant figures and enter your answer…

A: The expression given is 100sin65. To find- The decimal approximation of the given expression correct…

Q: Find the area and arc length 1) 11 ft 3) 315 16 ft

Q: A star rotates in a circular orbit about the center of its galaxy. The radius of the orbit is 3.7 x…

A: Please check step 2 for the solution.!

Q: If y = cose z then value of y' will be Select one: Az sine A O e-z, sine-Az O d.e dz¸ sine d O -A. e…

Q: Use double angle identities to find the indicated value. 17) cos e = sin 0 < 0 13 Find sin(20).

A: given cosθ=513we need to find the value sin2θ using double angle identities.

Q: tan? 0 8. -1= tan 0 sin? 0

Q: Find the amplitude, period and frequency of the simple harmonic motion of a particle moving…

A: We are given that a particle is moving uniformly around a circle of radius 3 units, with angular…

Q: Differentiate y = csc x(x + cot x).

Question

**Problem: Polar Curves and Tangents**

**Task:** Sketch the polar curve $ r^2 = \cos(6\theta) $ and compute the tangent line at the value $ \theta = \frac{\pi}{24} $.

**Instructions:**
1. Begin by understanding the polar equation $ r^2 = \cos(6\theta) $.
- This equation expresses a relationship between the radius $ r $ and the angle $ \theta $ in polar coordinates.
- It involves a cosine term with a multiple angle, which suggests the curve will have multiple symmetries or petals.

2. To sketch the curve:
- Evaluate $ r $ for several values of $ \theta $ ranging from 0 to $ 2\pi $.
- Note when $ \cos(6\theta) $ results in positive, zero, and negative values to determine where the curve crosses the pole (origin) and where it is undefined.

3. To find the tangent line at $ \theta = \frac{\pi}{24} $:
- Calculate the derivative of $ r $ with respect to $ \theta $ to find the slope of the tangent.
- Use the relationship $ \frac{dr}{d\theta} $ and evaluate it at $ \theta = \frac{\pi}{24} $.

4. Consider converting to Cartesian coordinates if necessary for finding the equation of the tangent line:
- Use the transformation $ x = r\cos(\theta) $ and $ y = r\sin(\theta) $.

5. Analyze graphically:
- Diagrams may illustrate how the curve behaves and where the tangent line is located for better visualization.

By understanding these steps, you can accurately sketch and analyze the polar curve and compute the required tangent line.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Step 3

VIEW

Step by step

Solved in 3 steps with 3 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Graphs$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

Similar questions

I will rate and like! Thank you!
5 arc sin y dy Answer with text and/or attachments:
Please show your work! How you get your works. I got the answers provided i just need to know steps by steps
Answer in radians pls
5) Let sinA= -5/13 with A in Quadrant 3 and find cot(2A)= 6) let cosA= 3/sqrt (10) with A in Quadrant 1 and find din(2A)= 7)let cosA= 1/sqrt(10) with A in Quadrant 4 and find cot(2A)= 8) let cosA= -1/sqrt(10) with A in Quadrant 3 and find cot(2A)=
9. Differentiate. Simplify where possible. y = arctan(cos 0)
Can someone please help? Thank you.
Determine the width of a petal of the curver= 2 cos(20) (The width is defined to be the widest part of the petal)