Name: Trigonometric Equation to Convert Cosine or Sine
Uploaded: 2019-07-19
Duration: 4 min 11 s
Description: Watch the video to learn how to solve trigonometric equations involving both sine and cosine simultaneously to find the general solution of the given equation.

Question

If

2 \cos x + \sin x = 1

, then find the sum of all possible values of

7 \cos x + 6 \sin x .

Accepted Answer

Given, $2 \cos x + \sin x = 1$

$\Rightarrow 2 \{\frac{1 - \tan^{2} (\frac{x}{2})}{1 + \tan^{2} (\frac{x}{2})}\} + \frac{2 \tan \frac{x}{2}}{1 + \tan^{2} \frac{x}{2}} = 1$

$\Rightarrow 2 - 2 \tan^{2} \frac{x}{2} + 2 \tan \frac{x}{2} = 1 + \tan^{2} \frac{x}{2}$

$\Rightarrow 3 \tan^{2} \frac{x}{2} - 2 \tan \frac{x}{2} - 1 = 0$

$\Rightarrow 3 \tan^{2} \frac{x}{2} - 3 \tan \frac{x}{2} + \tan \frac{x}{2} - 1 = 0$

$\Rightarrow 3 \tan^{2} \frac{x}{2} (\tan \frac{x}{2} - 1) + 1 (\tan \frac{x}{2} - 1) = 0$

$\Rightarrow (3 \tan \frac{x}{2} + 1) (\tan \frac{x}{2} - 1) = 0$

$∴ \tan \frac{x}{2} = - \frac{1}{3}, 1$

Case (1) : When $\tan \frac{x}{2} = - \frac{1}{3}$

$7 \cos x + 6 \sin x = 7 (\frac{1 - \tan^{2} \frac{x}{2}}{1 + \tan^{2} \frac{x}{2}}) + 6 (\frac{2 \tan \frac{x}{2}}{1 + \tan^{2} \frac{x}{2}})$

$= 7 (\frac{1 - \frac{1}{9}}{1 + \frac{1}{9}}) + 6 (\frac{- \frac{1}{9} \times 2}{1 + \frac{1}{9}})$

$= 7 \times \frac{8}{10} + (- 4) \cdot \frac{9}{10}$

$∴ 7 \cos x + 6 \sin x = 2 . . . . (1)$

Case (2) : When $\tan \frac{x}{2} = 1$

$7 \cos x + 6 \sin x = 7 (\frac{1 - \tan^{2} \frac{x}{2}}{1 + \tan^{2} \frac{x}{2}}) + 6 (\frac{2 \tan \frac{x}{2}}{1 + \tan^{2} \frac{x}{2}})$

$= 7 (\frac{1 - 1}{1 + 1}) + 6 (\frac{2 \times 1}{1 + 1}) = 6$

$∴$ Sum of all possible values of $7 \cos x + 6 \sin x$ is equal to $6$ .

If $2 \cos x + \sin x = 1$ , then find the sum of all possible values of $7 \cos x + 6 \sin x .$

Important Questions on Trigonometric Equations and Inequalities

Learn and Prepare for any exam you want !

If 2cosx+sinx=1, then find the sum of all possible values of 7cosx+6sinx.

Important Questions on Trigonometric Equations and Inequalities

Learn and Prepare for any exam you want !

If $2 \cos x + \sin x = 1$ , then find the sum of all possible values of $7 \cos x + 6 \sin x .$