If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

# Analyzing related rates problems: equations (trig)

Finding the rate of change of an angle that a falling ladder forms with the ground.

## Want to join the conversation?

• At , the results are presented in radians per minute. How do we know that the results are in radians and not some other unit of measure of angles (degrees perhaps)?

Typically it's not an issue because I would either input a figure measured in radians into a formula or tell my calculator to give me results in radians, however in this case all of the inputs are in meters and minutes and the rate of change of the angle just pops out in radians. At what point did we introduce radians, as the unit of measure of the angle, into the equation? •  That's right. When we say the derivative of cos(x) is -sin(x) we are assuming that "x" is in radians. In degrees it would be "(d/dx)cos(x) = -sin(x)(π/180)" because the "x" in degrees increases in a rate 180/π times faster than in radians.
• At , it says "if you take the hypotenuse times the cosine of theta, you would get x." I'm not sure how that was worked out. How does cosine relate to x? • I think I just broke all the rules using arccos to solve this problem. Are steps 1 and 3 I used valid? Is there a more concise way to use arccos that I overlooked? Thanks!
1. Because x’(t) = 3, let x(t) = 3t
2. x(t0)= 5sqrt(7) Pythagorean theorem
3. Combining steps 1 and 2, t0 becomes 5/3*sqrt(7) minutes
4. d/dt arccos(3t/20) = -3/sqrt(400 - 9*t^2)
5. Set t = 5/3 sqrt(7): -3/sqrt(400 - 9*(5/3 sqrt(7))^2) = -1/5 • You lost me at step 1. It appears you took the antiderivative of x'(t) = 3 to get x(t) = 3t. The antiderivative of x'(t) = 3 is actually 3t + C. From there, I understand step 2, but how did you get step 3, specifically? In what way did you combine steps 1 and 2? Step 4 looks like it would take an application of the chain rule, which I don't see evidence of. I didn't even look at step 5.

Can you please review how you got step 3 and 4, explicitly?
• Could this question be answered using tan instead of cos for the initial equation set up?

For instance, instead of x(t) = 20(cos(θ(t))),
could you do x(t) = 15/(tan(θ(t)))?

I was only able to get as far as x'(t) = 15(sec^2(θ(t))) * (θ'(t)) because when I simplified, one of the variables was x(t0) which was not one of the given constants. If someone can give me a step-by-step solution of this if possible that would be great • You can use tangent but 15 isn't a constant, it is the y-coordinate, which is changing so that should be y(t).
When you solve for θ you'll get
θ = arctan(y(t)/x(t))
then to get θ', you'd use the chain rule, and then the quotient rule.
During the quotient rule you'll get a y'(t), which isn't given, so then you'll have to set up another related rates equation between y and x to get y', and then plug that back in, etc.
It would take a lot lot more work.
You want to minimize the number of variables in the equation, so to avoid using x AND y, Sal used the relationship between variable x and the CONSTANT 20.
(1 vote)
• What does sub zero mean? • sub anything indicates a subscript, a small number on the lower right of a number, an exponent is a superscript, a small number or letter in the upper right.

subscripts are usually used to label things, especially in an order. so if you have multiple variables a lot of times people will label them like x sub 0, x sub 1, x sub 2 and so on. To write it out in typing it's usually done like this, x_0 with an underscore.

Since a lot of the time the ordering starts with 0, a variable sub zero is often used to say "starting point" so here t_0 is starting time.
• I just don't understand how at , the derivative turns into -20 sin theta. As far as I can tell, there was no -1 exponent to multiply onto the coeffecient of 20 and make it negative. Also how did the sin theta come in? • At - Sal says that x(t) = 20 * cos(theta),
What videos can I watch to study where he got that? My trigonometry is weak :( • Why radians?? when did they even come in • Radians were made by God (naturally occurring), degrees were made arbitrarily by man. Imagine the line that an angle traces along the unit circle's circumference.
What's the most natural way to describe the size of that angle? Well, it's the proportion of the ENTIRE circumference that the line drawn by the angle takes up.
God made the circumference of all the circles in the universe 2*pi*r right? So the unit circle has a 2*pi*(1) or just 2*pi circumference.
So the "angle" that draws a line that goes around half of that circumference is 1/2 of 2*pi or just pi; a quarter of the whole circumference is 1/4 of 2*pi or pi/2.  