Westonci.ca offers fast, accurate answers to your questions. Join our community and get the insights you need now. Discover the answers you need from a community of experts ready to help you with their knowledge and experience in various fields. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
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]
We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.
