Solving the Tricky 3cos(x) + sin(x) - 1 Problem

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In summary, the conversation was about a difficult trigonometry problem on a math final. The question was to find all positive values of x for a given equation, and the person had trouble solving it using various methods. The suggested approach was to use parametric formulae, which involves substituting sin(x) and cos(x) in terms of a new variable, t. This method is considered simpler and more effective for solving linear trigonometry equations.
  • #1
wasteofo2
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On my math final, there was this bastard of a trig problem that I simply couldn't solve. I knew the equations to use, how to solve the problem, but the answers just didn't work...

Anyway, the question was this:
Find all positive values of x for x being greater than or equal to zero, and less than or equal to 360.
3cos(x) + sin(x) - 1

How would you go about solving this? I tried graphing the equation and finding the x-intercept, but the values i got didn't work for whatever reason...
 
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  • #2
When you say 360, do you mean degrees?
Since cos(x)=sqrt(1-sin2(x)), you can set it up as a quadratic equation in cos(x), solve and go from there - discarding any solutions where sin2+cos2 does not add up to 1.
 
  • #3
The best way when you have a linear trig. equation is to use parametric formulae

t=tan(x/2)

then sin(x)=2t/(t^2+1) and cos(x)=(1-t^2)/(1+t^2)

you substitute them into the equation and solve it into t, simple, isn't it?

Never heard about that? Quite strange.
 

Related to Solving the Tricky 3cos(x) + sin(x) - 1 Problem

1. How do I approach solving a problem involving trigonometric functions?

The first step in solving a problem involving trigonometric functions is to identify and understand the given function. In this case, the given function is 3cos(x) + sin(x) - 1. Then, you can use any appropriate trigonometric identities or techniques to simplify the function and solve for the desired variable.

2. What is the most efficient way to solve the tricky 3cos(x) + sin(x) - 1 problem?

The most efficient way to solve this problem is to use the Pythagorean identity, sin^2(x) + cos^2(x) = 1, to rewrite the function as sin(x)(1 + 3cos(x)) - 1. This will allow you to group the terms and factor out sin(x) to solve for the variable.

3. Can I use a calculator to solve this problem?

Yes, you can use a calculator to solve this problem. Most scientific calculators have trigonometric functions and identities programmed into them, making it easier to solve complex trigonometric equations.

4. How do I check if my solution to the problem is correct?

You can check your solution by plugging your answer back into the original equation and simplifying. If both sides of the equation are equal, then your solution is correct.

5. Are there any common mistakes to avoid when solving trigonometric problems?

Some common mistakes to avoid when solving trigonometric problems include forgetting to convert degrees to radians, incorrectly applying trigonometric identities, and forgetting to check for extraneous solutions. It is important to carefully follow the steps and check your work to avoid these mistakes.

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