Really simple system of equations

[yoleven]

Homework Statement

1. 2500-Tac cos $$\Phi=0$$
2. Tac sin$$\Phi-500=0$$

Homework Equations

The Attempt at a Solution

I have 2 equations. I need to solve for $$\Phi$$ to get Tac. I have got to the point of :
Tac=500/sin$$\Phi$$
2500-500/sin$$\Phi*(cos\Phi)=0$$

after this i can't go any further

I am looking at the solution of 11.31°. But I need to solve to get it.

 

[tiny-tim]

I have got to the point of :
Tac=500/sin$$\Phi$$
2500-500/sin$$\Phi*(cos\Phi)=0$$

after this i can't go any further

I am looking at the solution of 11.31°. But I need to solve to get it.

Hi yoleven!

(have a phi: φ )

Your second line should be 2500 - 500cosφ/sinφ = 0.

 

[baba89]

As a scientist, it is important to understand and use mathematical equations to solve problems. In this case, we have a system of equations with two unknowns, Tac and \Phi. To solve for these variables, we can use substitution or elimination methods.

Substitution method:
From the first equation, we can express Tac in terms of \Phi as Tac = 2500/cos\Phi. Substituting this value into the second equation, we get 2500/cos\Phi * sin\Phi - 500 = 0. Simplifying, we get 2500*tan\Phi - 500 = 0. Solving for tan\Phi, we get tan\Phi = 500/2500 = 0.2. Using a calculator, we can find the inverse tangent of 0.2, which is approximately 11.31°.

Elimination method:
Multiplying the first equation by sin\Phi and the second equation by cos\Phi, we get 2500sin\Phi - Tac = 0 and Tacsin\Phi - 500cos\Phi = 0. Adding these two equations, we get 2500sin\Phi - 500cos\Phi = 0. Dividing both sides by 500, we get sin\Phi - cos\Phi = 0. Using the trigonometric identity sin^2\Phi + cos^2\Phi = 1, we can rewrite this equation as sin\Phi = cos\Phi. Taking the inverse sine of both sides, we get \Phi = 45°. However, we need to check if this is a valid solution by substituting it into the original equations. We can see that it does not satisfy the second equation. Therefore, we need to use the first method to find the correct solution of approximately 11.31°.

In conclusion, we can solve this system of equations using different methods and obtain the same solution of approximately 11.31° for \Phi. This shows the importance of understanding and using mathematical equations in scientific research.