# Trigonometric Equation with Multiple Functions and Arguments

• Islwyn
In summary, the conversation discusses solving an equation with multiple trig functions and using identities to express everything in terms of cosine in order to get a polynomial equation. However, it is unlikely to find a closed form expression for 'a', but numerical values can be used to approximate solutions.
Islwyn
Can anyone help solve the below for 'a'

0=b(ycos(a)-bcos(2a)+xsin(a))

I've reduced it to

b=ycos(a)+2sin(a)sin(a)+xsin(a)

=sin(a)(cot(a)+2sin(a)+x)

Which I think is right, but I can't get any further without just ending up with different forms that appear more complex.

What makes this equation tough is the fact that you've got 2 different trig functions running around there. What's more you've got 2 different arguments in your cosine functions. I would try to express everything in terms of $\cos(a)$, so that you end up with a polynomial equation in that quantity.

Try to make use of the following identities:

$$\cos(2a)=2\cos^2(a)-1$$

$$\sin(a)=\sqrt{1-\cos^2(a)}$$

I don't think you're going to get a closed form expression for a, but if you have numerical values for the parameters you will be able to get approximate solutions.

## What is a trigonometric equation?

A trigonometric equation is an equation that contains trigonometric functions such as sine, cosine, tangent, etc. These equations involve finding the values of the unknown variables that satisfy the equation.

## What are the steps to solve a trigonometric equation?

The steps to solve a trigonometric equation are:

1. Isolate the trigonometric function on one side of the equation.
2. Use inverse trigonometric functions to eliminate the trigonometric function and find the angle.
3. Solve for the angle using algebraic techniques.
4. Check the solution by substituting the value of the angle back into the original equation.

## What are the common trigonometric identities used to solve equations?

The most commonly used trigonometric identities to solve equations are:

• Pythagorean identities: sin^2(x) + cos^2(x) = 1 and tan^2(x) + 1 = sec^2(x)
• Reciprocal identities: sin(x) = 1/csc(x), cos(x) = 1/sec(x), and tan(x) = 1/cot(x)
• Double angle identities: sin(2x) = 2sin(x)cos(x), cos(2x) = cos^2(x) - sin^2(x), and tan(2x) = 2tan(x)/1-tan^2(x)

## Can trigonometric equations have more than one solution?

Yes, trigonometric equations can have an infinite number of solutions. This is because the trigonometric functions are periodic and have repeating patterns. Therefore, the solutions can be found for any value of the angle within the given range.

## How can I check if my solution is correct?

To check if your solution is correct, you can substitute the value of the angle back into the original equation and see if the equation holds true. You can also use a calculator to evaluate the trigonometric function and compare it to the value obtained from solving the equation.

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