How to solve this system of equations of trig functions

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• ual8658
In summary: So the final solutions are t=1.155/( sin(\alpha)) and t= 1.155/( sin(\alpha)) + cos(\alpha)/(sin(\alpha))In summary, the conversation discussed solving a quadratic equation with two unknowns using various methods, including substitution and the double-angle formulas. The final solutions were determined to be t=1.155/( sin(\alpha)) and t= 1.155/( sin(\alpha)) + cos(\alpha)/(sin(\alpha)).
ual8658

I've written it out and it seems impossible. I get -50(sin^2(alpha)) = 86.63 cos(alpha) sin(alpha) - 6.54. Where would I go from there?

Assuming the first term is ## (75 cos(\alpha))t ## and not ## 75 cos(\alpha t) ##, the equation is a quadratic in t, and has a completely separate solution from ## t=1.155/( sin(\alpha)) ##. If the second case is indeed what is asked for, the solution is non-trivial, and a numeric solution, e.g. by graphing a couple of the terms and finding where they intersect, would be in order.

Logical Dog
There is an analytic solution. Based on your current progress: Replace the cosine using the sine, then simplify. You'll get a quadratic equation to solve (with a small trick involved).

ual8658 said:
I've written it out and it seems impossible. I get -50(sin^2(alpha)) = 86.63 cos(alpha) sin(alpha) - 6.54. Where would I go from there?

Well, you can rewrite it as:

$86.63 cos(\alpha) sin(\alpha) = 6.54 - 50(sin^2(\alpha))$

You can rewrite $cos(\alpha) = \sqrt{1-sin^2(\alpha)}$ to get an equation only involving $sin(\alpha)$. Alternatively, you could use the double-angle formulas:

$sin(2 \alpha) = 2 sin(\alpha) cos(\alpha)$
$cos(2 \alpha) = 1 - 2 sin^2(\alpha)$

My mistake, I overlooked the solution that @mfb is referring to. It's essentially two equations and two unknowns.

ual8658
mfb said:
There is an analytic solution. Based on your current progress: Replace the cosine using the sine, then simplify. You'll get a quadratic equation to solve (with a small trick involved).

This is going way over my head. Could you elaborate?

stevendaryl said:
Well, you can rewrite it as:

$86.63 cos(\alpha) sin(\alpha) = 6.54 - 50(sin^2(\alpha))$

You can rewrite $cos(\alpha) = \sqrt{1-sin^2(\alpha)}$ to get an equation only involving $sin(\alpha)$. Alternatively, you could use the double-angle formulas:

$sin(2 \alpha) = 2 sin(\alpha) cos(\alpha)$
$cos(2 \alpha) = 1 - 2 sin^2(\alpha)$

I tried using the double angle formula with sin(2 alpha) but I'm still stuck after that.

When you substitute with ## cos(\alpha)=\sqrt{1-sin^2(\alpha)} ## , you then square both sides of the equation and get a quadratic equation in ## u=sin^2(\alpha) ##. (It is actually 4th power in ## sin(\alpha) ## , but there is no ## sin^3(\alpha) ## term and no ## sin(\alpha) ## term=it is quadratic in ## sin^2(\alpha) ##.

When you substitute with ## cos(\alpha)=\sqrt{1-sin^2(\alpha)} ## , you then square both sides of the equation and get a quadratic equation in ## u=sin^2(\alpha) ##. (It is actually 4th power in ## sin(\alpha) ## , but there is no ## sin^3(\alpha) ## term and no ## sin(\alpha) ## term=it is quadratic in ## sin^2(\alpha) ##.

Thank you! I got it now.

1. How do I solve a system of equations involving trigonometric functions?

Solving a system of equations involving trigonometric functions typically involves using algebraic techniques and trigonometric identities to eliminate variables and find a solution. It is important to first rewrite the equations in terms of a common variable, such as sine or cosine, and then use substitution or elimination to solve for the remaining variables.

2. What are some common trigonometric identities used to solve systems of equations?

Some common trigonometric identities used to solve systems of equations include the Pythagorean identities (sin^2x + cos^2x = 1), the sum and difference identities (sin(x+y) = sinxcosy + cosxsiny, sin(x-y) = sinxcosy - cosxsiny), and the double angle identities (sin2x = 2sinxcosx, cos2x = cos^2x - sin^2x).

3. Can I use a graphing calculator to solve a system of equations involving trig functions?

While a graphing calculator can be helpful in visualizing the solutions to a system of equations, it is not recommended to solely rely on it for finding the exact solutions. It is important to also understand the algebraic methods used to solve the equations in order to ensure the accuracy of the solutions.

4. Are there any specific steps or strategies for solving systems of trigonometric equations?

One common strategy for solving systems of trigonometric equations is to use the substitution method. This involves solving one equation for a variable in terms of the other, and then substituting that expression into the other equation. Another strategy is to use the elimination method, which involves manipulating the equations to eliminate a variable and solve for the remaining variables.

5. Are there any real-world applications for solving systems of trigonometric equations?

Systems of trigonometric equations can be used in various fields such as engineering, physics, and navigation. For example, in engineering, these equations can be used to model the behavior of oscillating systems, while in navigation, they can be used to determine the position and trajectory of objects in space.

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