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Sagot :
Answer:
F. 30° + 360°n, 150° + 360°n, 210° + 360°n, 330° + 360°n
Step-by-step explanation:
Given trigonometric equation:
[tex]4\cos^2(x)=3[/tex]
To solve the given trigonometric equation for all x, begin by dividing both sides by 4 to isolate cos²(x):
[tex]\cos^2(x)=\dfrac{3}{4}[/tex]
Now, square root both sides:
[tex]\cos(x)=\pm\sqrt{\dfrac{3}{4}}=\pm\dfrac{\sqrt{3}}{2}[/tex]
According to the unit circle, the cosine of an angle is equal to √3/2 when x = 30° and x = 330°, and the cosine of an angle is equal to -√3/2 when x = 150° and x = 210°.
Since cosine is a periodic function with a period of 360°, it repeats its values every 360°. Therefore, when solving trigonometric equations involving cosine, we need to consider adding multiples of 360° to each solution to account for all possible solutions within the given range.
So, the solutions of the given trigonometric equation are:
[tex]\Large\boxed{\begin{array}{l}30^{\circ} + 360^{\circ}n\\150^{\circ} + 360^{\circ}n\\210^{\circ} + 360^{\circ}n\\330^{\circ} + 360^{\circ}n\end{array}}[/tex]
where n is an integer.
[tex]Tosolve the equation\( 4 \cos^2(x) = 3 \) for all \( x \), we can follow these steps:### Steps to Solve the Equation1. **Isolate \( \cos^2(x) \):** \[ 4 \cos^2(x) = 3 \] Divide both sides by 4: \[ \cos^2(x) = \frac{3}{4} \][/tex][tex]2. **Take the square root of both sides:** \[ \cos(x) = \pm \sqrt{\frac{3}{4}} = \pm \frac{\sqrt{3}}{2} \][/tex][tex]3. **Determine the angles \( x \) where \( \cos(x) = \pm \frac{\sqrt{3}}{2} \):** - The cosine function is positive or negative at specific standard angles. Here are the relevant angles: - \( \cos(x) = \frac{\sqrt{3}}{2} \) at \( x = 30^\circ \) and \( x = 330^\circ \) - \( \cos(x) = -\frac{\sqrt{3}}{2} \) at \( x = 150^\circ \) and \( x = 210^\circ \)[/tex][tex]4. **Express the solutions in general form:** - Since the cosine function has a period of \( 360^\circ \), we add multiples of \( 360^\circ \) (denoted by \( n \) where \( n \) is an integer) to each solution: \[ x = 30^\circ + 360^\circ n \] \[ x = 150^\circ + 360^\circ n \] \[ x = 210^\circ + 360^\circ n \] \[ x = 330^\circ + 360^\circ n \][/tex]Answer
[tex]**F.** \( 30^\circ + 360^\circ n, 150^\circ + 360^\circ n, 210^\circ + 360^\circ n, 330^\circ + 360^\circ n \)This option lists all the possible solutions for \( x \) where \( \cos(x) = \pm \frac{\sqrt{3}}{2} \).[/tex]
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