Solve the trigonometric equation involve sin(x), cos(x) and sin(x)cos(x)

• daphnelee-mh
In summary, the conversation discusses the difficulty in finding an angle, with two possible numerical solutions given. One of the solutions is confirmed to be correct and it is suggested that a numerical solution is more practical than an analytical solution in this case. The conversation also addresses a side note on the left-hand side of an equation and provides additional help in the form of a trigonometric identity.

daphnelee-mh

Homework Statement
I had problem with solving the equation while doing my mechanics homework

How to solve equation : 1226.25cosx - 15000sinxcosx + 7500sinx = 0
Relevant Equations
sin^2 + cos^2 = 1

I can’t get the angle, answer given is x=56.33 , x=9.545. (All steps before the equation are correct.)

daphnelee-mh said:
I can’t get the angle, answer given is x=56.33 , x=9.545.
You mean ##\theta = 0.983074## or ##\theta = 56.326^\circ## . Not sure what the x=9.545 means.
All steps before the equation are correct
Confirmed.
Aren't you in a situation where a numerical answer is adequate ? I don't think an analytical answer is possible.

Side note: LHS of your last equation should be a cotangent.

etotheipi
I tried letting ##u = \cos{\theta}## and ended up with a quartic in ##u##, which my calculator could solve (I think it uses the quartic formula), but I certainly wouldn't have the patience to do that myself!

BvU said:
You mean ##\theta = 0.983074## or ##\theta = 56.326^\circ## . Not sure what the x=9.545 means.
Confirmed.
Aren't you in a situation where a numerical answer is adequate ? I don't think an analytical answer is possible.

Side note: LHS of your last equation should be a cotangent.

Hi daphne:

The following may be of some additional help.

sin 2x = 2 sin x cos x

Regards,
Buzz

daphnelee-mh said:
I suppose this is a reply to my
BvU said:
Not sure what the x=9.545 means

And I must humbly admit I discarded the second answer in the wolfram link because it's so far away from the initial 60##^\circ##. But it satisfies the equation.

Is it clear that an analytical solution is unlikely to be found for this problem, so a numerical solution should be acceptable ?

1. What is a trigonometric equation?

A trigonometric equation is an equation that involves trigonometric functions, such as sine, cosine, tangent, etc. These functions represent the relationship between the sides and angles of a right triangle.

2. How do I solve a trigonometric equation?

To solve a trigonometric equation, you must use algebraic techniques to isolate the variable and then use trigonometric identities and properties to simplify the equation. You may also need to use your knowledge of inverse trigonometric functions to find the solutions.

3. What is the difference between solving a trigonometric equation and finding a trigonometric identity?

Solving a trigonometric equation means finding the values of the variable that make the equation true. On the other hand, finding a trigonometric identity means proving that two expressions are equal for all values of the variable.

4. Can a trigonometric equation have more than one solution?

Yes, a trigonometric equation can have multiple solutions. This is because trigonometric functions are periodic, meaning they repeat their values at regular intervals. Therefore, there may be more than one value of the variable that satisfies the equation.

5. How do I check if my solution to a trigonometric equation is correct?

You can check your solution by substituting it back into the original equation and simplifying to see if both sides are equal. You can also use a graphing calculator to plot the original equation and your solution to visually confirm if they intersect at the same point.