Solving Trig Related Physics Problem: Step by Step Guide

[rockytriton]

I'm trying to solve a physics problem with some trig in it. I'm looking at a solution but I'm not understanding where it gets from one step to another.

I understand it up to this point:

19.6m + 800m * cos(Z)^2 = 2,000m * sqr(1 - cos(Z)^2) * cos(Z)

But I don't understand how they get from that to:

384m^2 + 31,360m^2 * cos(Z)^2 + 640,000m^2 * cos(Z)^4 =
4,000,000m^2 * cos(Z)^2 - 4,000,000m^2 * cos(Z)^4

Is this some kind of crazy trig trick that I don't know about?

 

[metacristi]

I'm trying to solve a physics problem with some trig in it. I'm looking at a solution but I'm not understanding where it gets from one step to another.

I understand it up to this point:

19.6m + 800m * cos(Z)^2 = 2,000m * sqr(1 - cos(Z)^2) * cos(Z)

But I don't understand how they get from that to:

384m^2 + 31,360m^2 * cos(Z)^2 + 640,000m^2 * cos(Z)^4 =
4,000,000m^2 * cos(Z)^2 - 4,000,000m^2 * cos(Z)^4

Is this some kind of crazy trig trick that I don't know about?

Raise both sides of the equation to the second power and here you are (approximately, should have been 384.16m^2).

 

[tacman]

Solving trigonometry related physics problems can definitely be challenging, but with a step by step guide, it can become much easier. Let's break down the solution to this problem and hopefully it will become clearer to you.

First, we can start by simplifying the left side of the equation by using the distributive property. We can multiply 800m by both terms inside the parentheses to get:

19.6m + 640,000m * cos(Z)^2 = 2,000m * sqr(1 - cos(Z)^2) * cos(Z)

Next, we can use the trigonometric identity cos^2(Z) + sin^2(Z) = 1 to simplify the term inside the square root on the right side of the equation. This gives us:

19.6m + 640,000m * cos(Z)^2 = 2,000m * sqr(sin^2(Z)) * cos(Z)

We can then simplify the square root by taking the square root of sin^2(Z), which is just sin(Z). This gives us:

19.6m + 640,000m * cos(Z)^2 = 2,000m * sin(Z) * cos(Z)

Next, we can use the trigonometric identity sin(Z) * cos(Z) = sin(2Z) to further simplify the right side of the equation. This gives us:

19.6m + 640,000m * cos(Z)^2 = 2,000m * sin(2Z)

Now, we can use the trigonometric identity sin(2Z) = 2sin(Z) * cos(Z) to get rid of the sin(2Z) term on the right side of the equation. This gives us:

19.6m + 640,000m * cos(Z)^2 = 4,000m * sin(Z) * cos(Z)

Finally, we can use the trigonometric identity cos^2(Z) = 1 - sin^2(Z) to substitute for cos(Z)^2 on the left side of the equation. This gives us the final result:

19.6m + 640,000m * (1 - sin^2(Z)) = 4,000m * sin(Z) * cos(Z)

We can then use algebra to expand and simplify the left side of the equation, which leads to the equation you mentioned:

384m^2 +