Derivatives of Trig with Triangles

In summary, the conversation discusses finding the rate at which the area of a triangle is decreasing when the angle between two sides of fixed length changes. The given information includes the lengths of the two sides, the rate of change of the angle, and the desired angle. The use of Heron's formula is questioned, and it is suggested to express the area as a function of the given information. The conversation ends with the solution being found and a clarification on taking the derivative of a trigonometric function.
  • #1
LadiesMan
96
0
[SOLVED] Derivatives of Trig with Triangles

1. Two sides of a triangle are six and eight metres in length. If the angle between them decreases at the rate of 0.035 rad/s, find the rate at which the area is decreasing when the angle between the sides of fixed length is pi/6. Answer 0.727 m^2/min


2. I tried using Heron's formula but I realized it was almost impossible to take the derivative somehoe with the angle needed. My givens are
dtheta/dt = -0.035 rad/s
the angle between them is pi/6 or 30 degrees
Now to find da/dt...

Thanks
 
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  • #2
I'm not sure why you're using Heron's formula - that looks to be better when you know all three sides and don't necessarily know any of the angles.

Can you express the area as a function of the two fixed lengths and the contained angle? It's pretty simple to take the derivative of the relevant expression.
 
  • #3
no sry i don't understand how to express the equation but what i think is cut the triangle into two?
 
  • #4
nvmi got it thank you very much! =) But one thing when u take the derivative of 24sin theta,w hy is it that u leave the 24?
 
  • #5
I'm not sure I understand your question. It's just:

d/dt (k*f(t)) = k* df/dt , k = constant

Is that clear?
 

1. What are the basic trigonometric functions and their derivatives?

The basic trigonometric functions are sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc). Their derivatives are as follows:

  • sin x: cos x
  • cos x: -sin x
  • tan x: sec^2 x
  • cot x: -csc^2 x
  • sec x: sec x * tan x
  • csc x: -csc x * cot x

2. How do you find the derivative of a trigonometric function using the quotient rule?

The quotient rule states that the derivative of a function f(x) divided by g(x) is equal to (f'(x) * g(x) - f(x) * g'(x)) / (g(x))^2. To find the derivative of a trigonometric function using the quotient rule, first rewrite the function as a quotient of two functions. Then, use the quotient rule to find the derivative of each function separately. Finally, substitute the values back into the original function to find the derivative.

3. What is the chain rule and how is it used to find the derivative of a trigonometric function?

The chain rule is a method used to find the derivative of a composite function. In other words, it is used when a function is composed of two or more functions. To find the derivative of a trigonometric function using the chain rule, first identify the inner and outer functions. Then, use the chain rule formula: f'(g(x)) * g'(x), where f'(g(x)) is the derivative of the outer function evaluated at the inner function, and g'(x) is the derivative of the inner function.

4. How do you find the derivative of a trigonometric function using the product rule?

The product rule states that the derivative of a function f(x) multiplied by g(x) is equal to f'(x) * g(x) + f(x) * g'(x). To find the derivative of a trigonometric function using the product rule, first identify the two functions being multiplied together. Then, use the product rule formula to find the derivative of each function separately. Finally, add the two derivatives together to get the final result.

5. Can the derivatives of trigonometric functions be used to find the slope of a curve?

Yes, the derivatives of trigonometric functions can be used to find the slope of a curve. The slope of a curve at a specific point is equal to the derivative of that curve at that point. This can be useful in applications such as physics and engineering, where the slope of a curve represents the rate of change of a variable with respect to another variable.

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