Solving Trig Equations: Help Needed!

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In summary, the two solutions $240\sin(30^{\circ})+275\sin(40.9)=300$ and $59.86\sin(30^{\circ})+275\sin(79.13)=300$ are both valid solutions for the problem of equilibrium of a particle.
  • #1
Drain Brain
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Can you help me how to solve this system of trig eqns

$W\cos(30^{\circ})-275\cos(\theta)=0$
$W\sin(30^{\circ})+275\sin(\theta)=300$

I have tried to divide the first eqn by 2nd and I get

$\tan(30^{\circ})=\frac{275}{300\cos(\theta)}-\tan(\theta)$ I'm stuck here! Kindly help me please!
 
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  • #2
Drain Brain said:
Can you help me how to solve this system of trig eqns

$W\cos(30^{\circ})-275\cos(\theta)=0$
$W\sin(30^{\circ})+275\sin(\theta)=300$

I have tried to divide the first eqn by 2nd and I get

$\tan(30^{\circ})=\frac{275}{300\cos(\theta)}-\tan(\theta)$ I'm stuck here! Kindly help me please!

Hey Drain Brain!

Suppose we isolate $\cos(\theta)$ and $\sin(\theta)$ in each equation, squared them, and add them.
Then we would be rid of $\theta$ and can solve for $W$.

Afterwards, we can substitute $W$ back in the first equation and solve for $\theta$.
Finally, we should check if the solutions found are actually solutions, since we may have introduced new solutions.
 
  • #3
HI I LIKE YOU!:eek:

I just did what said and came up with a quadratic equation

$W^2-600W\sin(30)+300^2-275^2=0$

the two solutions are

$W=240.1$ and $W=59.86$

$\theta = 40.9$ and $\theta = 79.13$

which of them should I choose?
 
  • #4
Drain Brain said:
HI I LIKE YOU!:eek:

I just did what said and came up with a quadratic equation

$W^2-600W\sin(30)+300^2-275^2=0$

the two solutions are

$W=240.1$ and $W=59.86$

Good! ;)
$\theta = 40.9$ and $\theta = 79.13$

which of them should I choose?

Actually, you should have 2 solutions for $\theta$ for each value of $W$...

What happens if we substitute them in the second equation?
 
  • #5
I like Serena said:
Good! ;)

Actually, you should have 2 solutions for $\theta$ for each value of $W$...

What happens if we substitute them in the second equation?

$240\sin(30^{\circ})+275\sin(40.9)=300$-->>$300=300$ $59.86\sin(30^{\circ})+275\sin(79.13)=300$---->>$300=300$

Does this mean that the values of W and $\theta$ that I get are valid solutions?

Actually the system of equations that I posted above came from a problem about equilibrium of a particle.
I was asked to find force W and angle $\theta$ to satisfy equilibrium conditions. The answer to this problem was 240 lb for W, and $\theta=40.9$. If both sets of solution are valid, why the other solution was not chosen?
 
  • #6
Drain Brain said:
$240\sin(30^{\circ})+275\sin(40.9)=300$-->>$300=300$ $59.86\sin(30^{\circ})+275\sin(79.13)=300$---->>$300=300$

Does this mean that the values of W and $\theta$ that I get are valid solutions?

Actually the system of equations that I posted above came from a problem about equilibrium of a particle.
I was asked to find force W and angle $\theta$ to satisfy equilibrium conditions. The answer to this problem was 240 lb for W, and $\theta=40.9$. If both sets of solution are valid, why the other solution was not chosen?

Yes, they are both solutions.
It depends on the actual problem statement what to do with them.
Perhaps just a single solution was requested, perhaps there is another reason to discard one of them, or perhaps the given answer is simply incomplete.
 

FAQ: Solving Trig Equations: Help Needed!

What is a trigonometric equation?

A trigonometric equation is an equation that involves trigonometric functions, such as sine, cosine, and tangent. These equations often involve angles and can be used to solve for unknown values in a triangle or other geometric shape.

How do you solve a trigonometric equation?

To solve a trigonometric equation, you must use algebraic techniques to isolate the trigonometric function and then use inverse trigonometric functions to find the angle measure. This may involve using identities, factoring, or applying the unit circle.

What are the steps to solve a trigonometric equation?

The general steps to solve a trigonometric equation are:
1. Simplify the equation by using trigonometric identities if necessary.
2. Isolate the trigonometric function on one side of the equation.
3. Use inverse trigonometric functions to find the angle measure.
4. Check your solution by plugging it back into the original equation.

Why are trigonometric equations important?

Trigonometric equations are important because they are used to solve real-world problems involving angles and distances. They are also essential in fields such as engineering, physics, and astronomy.

What are some common mistakes when solving trigonometric equations?

Some common mistakes when solving trigonometric equations include:
- Forgetting to check for extraneous solutions
- Making errors while simplifying the equation
- Forgetting to use the unit circle or other trigonometric identities
- Incorrectly applying inverse trigonometric functions

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