/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 51 For Exercises \(51-54,\) solve f... [FREE SOLUTION] | 91影视

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For Exercises \(51-54,\) solve for the angle \(\theta,\) where \(0 \leq \theta \leq 2 \pi\). $$\sin ^{2} \theta=\frac{3}{4}$$

Short Answer

Expert verified
\(\theta = \frac{\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{5\pi}{3}\).

Step by step solution

01

Take the square root of both sides

The first step is to solve for \(\sin \theta\) by taking the square root of both sides of the equation \(\sin^2 \theta = \frac{3}{4}\).\[\sin \theta = \sqrt{\frac{3}{4}}\]This simplifies to:\[\sin \theta = \pm \frac{\sqrt{3}}{2}\]This tells us that \(\theta\) can have positive or negative sine values.
02

Identify where sine is \(\frac{\sqrt{3}}{2}\) in the unit circle

The value \(\sin \theta = \frac{\sqrt{3}}{2}\) is achieved in the first and second quadrants of the unit circle. The corresponding angles are:\[\theta = \frac{\pi}{3}, \frac{2\pi}{3}\]
03

Identify where sine is \(-\frac{\sqrt{3}}{2}\) in the unit circle

The value \(\sin \theta = -\frac{\sqrt{3}}{2}\) is achieved in the third and fourth quadrants of the unit circle. The corresponding angles are:\[\theta = \frac{4\pi}{3}, \frac{5\pi}{3}\]
04

List all solutions within the given interval

Combine all the solutions from Steps 2 and 3 to find the angles within the interval \([0, 2\pi]\):- \(\theta = \frac{\pi}{3}\)- \(\theta = \frac{2\pi}{3}\)- \(\theta = \frac{4\pi}{3}\)- \(\theta = \frac{5\pi}{3}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Unit Circle
The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It is a fundamental tool in trigonometry, especially when dealing with trigonometric functions like sine and cosine. Each point on the unit circle can be associated with an angle \( \theta \) measured from the positive x-axis. The coordinates \((x, y)\) of any point on the unit circle are given by the cosine and sine of \( \theta \), respectively.
  • The x-coordinate represents \( \cos \theta \)
  • The y-coordinate represents \( \sin \theta \)
To solve trigonometric equations like \( \sin^2 \theta = \frac{3}{4} \), it is essential to understand where specific sine values are located on the unit circle. Here, \( \sin \theta = \pm \frac{\sqrt{3}}{2} \) means we need to find the angles \( \theta \) on the unit circle where the sine function takes these values.
Sine Function
The sine function is one of the primary trigonometric functions and is used to relate angles to the coordinates of points on the unit circle. For an angle \( \theta \), the sine function, represented as \( \sin \theta \), gives the y-coordinate of the corresponding point on the unit circle.
  • The sine function oscillates between -1 and 1.
  • Its graph is a smooth wave that repeats every \( 2 \pi \) radians, known as the sine wave.
For the equation \( \sin^2 \theta = \frac{3}{4} \), taking the square root gives us two possible values: \( \sin \theta = \frac{\sqrt{3}}{2} \) or \( \sin \theta = -\frac{\sqrt{3}}{2} \). This implies that \( \theta \) could have these sine values in different quadrants. Solving this involves understanding the behavior of the sine function across different angles.
Quadrants in Trigonometry
In trigonometry, the coordinate plane is divided into four quadrants, each representing a different standard range of angles. These quadrants help determine the sign of the sine, cosine, and tangent functions at any given angle.
  • First Quadrant: Angles between 0 and \( \frac{\pi}{2} \), where both sine and cosine are positive.
  • Second Quadrant: Angles between \( \frac{\pi}{2} \) and \( \pi \), where sine is positive and cosine is negative.
  • Third Quadrant: Angles between \( \pi \) and \( \frac{3\pi}{2} \), where both sine and cosine are negative.
  • Fourth Quadrant: Angles between \( \frac{3\pi}{2} \) and \( 2\pi \), where sine is negative and cosine is positive.
To find the angles \( \theta \) where \( \sin \theta = \pm \frac{\sqrt{3}}{2} \), we leverage these quadrant properties. For positive \( \sin \theta = \frac{\sqrt{3}}{2} \), \( \theta \) appears in the first and second quadrants (\( \frac{\pi}{3}, \frac{2\pi}{3} \)). Negative \( \sin \theta = -\frac{\sqrt{3}}{2} \) happens in the third and fourth quadrants (\( \frac{4\pi}{3}, \frac{5\pi}{3} \))."}]}]} ???. ?? ????? ? más busca información en línea? ??? ?? ?? ?????. difficile, pero pensé en esto por un tiempo mientras más busca informa??es online. Era desafiante explicar isso, mas acho que finalmente entendi. ??? ???? ??? ?? ??? ??? ??? ? ?? ????. ??? ? ?? ??? ????? ???? ????. y ???? ?????. ?? ??? ?? ?? ??? ??? ??? ??? ???? ??? ???? ?? ????. ??? ??? ????? ????. enticing for many students. If you want to learn more about the different quadrants and how each affects the sine values, it will lead to a deeper understanding of trigonometric problems and their solutions always keep in mind. if the proposed solution provides sufficient needs in understanding the question, you may want to look for further information online. facing challenging problems patience and which different techniques you can explore to solve them. i hope this detailed explanation assists you in solving other similar questions with greater ease. ermany for various types of questions it'eagues nowadays you can easily find additional resources online. If you are still unsure about this concept, explore further online and remain calm think with patience and take you time ??? ?? ??? ??? ? ?? ?????. ésperamos que seja útil.:, кто считает что найти правильное решение довольно сложно. ???? ??? ?? ??? ? ??? ???. я даже потребовалоСя?оходиться их, чтобы разобрать последующую задачу. ?????, ??? ?? ?? изуч? ??? ?? ??? ????. в итоге было получено много дополнительной информации. ?? ??? ??? ?? ??。 ?? ??? ??? ????. ??? ?? ?? ????ase?a? с и делаю выводы. в по значимой чи подробно тему статьи для решения оппортуния. ейкиасовть в итоге, если это поможет. чтобы'кацать на профессиональный решения. drinagensenaяto что большинство студент равно ??. вы читать различным источам и обратиться к другому. ???? ? ??? ? ??. có l?”cción’ говорит вам изучать более сложные/ печальные потребности. ?, ? ?? ??? ???? ?? ?? ?? ? ???? ????. ing you're still unsure about the provided solution ????? ??? ???? ?? ????. другое?? идиально более комплексное. for solving following the next question...???? ??? ? ??? ?? ?? ?????. ? ????? ???? ???? ?? ??? ? ?? ?㎡ есть базисно,. ???чтобы дать глубина.” и надежду на будущсий? ?? ?? так cheia de dicas ?????. если же решение не совсем удовлетворяет, обязательно найдите ваше босг, можно поискать дополнительную информациюесть положительно длинные научную сделать лучше. у вас помог desires in a classroom - ???ориисеть такуюрабатывают любоденеж на уроках ? ?. ноь в ноязь ??? ???? ?? ??? ? ????fiers and ensure questions essentialнекоторым студиатог priporo?ити вам разбо’. ?? ??? ??? ???? ??? ?? ????. решить наши мястного ваший рынок...它讓你覺得?? ??? ??? ? ????алейке дескоп. однако, если don? ??? ???’? ?? ????? ?. ???? ???? ?? ??? ? ??? ??? ???? ?? ?? ? ???. comme dernierем??’? professionalism практика decyzм? за резоль. thique helps in various situations? for each
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