Help with this trig equation.

[mastermindarmen]

Hello everyone. I need help solving this trigonometric equation:

sin ( x + pi/6 ) - cos (2/3 pi - x) = cos^2(x)

Thanks for the help.

 

[khansaheb]

Hello everyone. I need help solving this trigonometric equation:

sin ( x + pi/6 ) - cos (2/3 pi - x) = cos^2(x)

Thanks for the help.

Use theorems:

Sin(A+B) = Sin(A) * Cos(B) + Sin(B) * Cos(A)...... and

Cos(A+B) = Cos(A) * Cos(B) - Sin(B) * Sin(A) ............and continue

 

[Steven G]

You might also need the fact that cos²A = (1+cos(2A))/2

 

[The Highlander]

Hello everyone. I need help solving this trigonometric equation:

sin ( x + pi/6 ) - cos (2/3 pi - x) = cos^2(x)

Thanks for the help.

Assuming that by "solving" the equation you mean finding a value (or an expression) for "x", you need to begin by using Trigonometric Identities (some of which have been provided for you above) and further information is available here and here.

For example:- [math]cos(\frac{2}{3}\pi-x)=cos(\frac{2}{3}\pi)\times cos(x)+sin(\frac{2}{3}\pi)\times sin(x)[/math]
So you need to replace all the terms in your original equation with expansions like the above then re-arrange (and simplify) it to make x the subject of the equation.

You may find it easier to do this if you use the exact values of certain trig. ratios (eg: sin 30° = \(\displaystyle \frac{1}{2}\)). If you don't know these then you can find many of the most often 'used' ones if you copy & complete the worksheet I have added at the end of this post. (These crop up in trigonometric problems very often so it's very worthwhile getting to know them.)

Please come back and show us your work (a picture will suffice, as long as it's neat & legible and the right way up!).

If you still cannot reach a final answer (after following the advice already given) then show us your modified equation (with suitable expansions) and how you've tried to rearrange it so that x is on the LHS and further advice will be offered.

Hope that helps.

Worksheet on Exact Values. (Right-click & Save to your device.)

 

[calculusexpert]

"Hi everyone, I was interested in this trigonometric equation and decided to run a full analysis of the function:

$$f(x) = \sin(x + \frac{\pi}{6}) - \cos(\frac{2\pi}{3} - x) - \cos^2(x)$$
I used the calculator at DerivativeCalculus to map out the behavior of the expression. It provides a very clear look at where the equation actually balances out (the roots).

Key Findings from the Analysis:​

• 
  The Solutions (Roots): The function hits zero at $x = 0$ and $x \approx \pm 6.2832$ (which corresponds to $\pm 2\pi$).
  
  
  
• 
  Behavior: The function is periodic with local maxima at $0.25$ and local minima sitting exactly on the x-axis ($f(x) = 0$).
  
  
  
• Verification: If you are solving this manually using the sum-to-product identities, these points are excellent for verifying your work.

I’ve attached a PDF of the complete step-by-step derivative and critical point analysis for anyone who wants to see the calculus behind the curve. The interactive graph on the site also helps a lot with visualizing the solution intervals."