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Chapter 4: Problem 18

Find all real numbers that satisfy each equation. $$ \sin x=\sqrt{3} / 2 $$

Short Answer

Expert verified
x = \( \frac{\pi}{3} + 2k\pi \) or \( \frac{2\pi}{3} + 2k\pi \)

Step by step solution

01

Recall the Unit Circle Values

Identify the angle values from the unit circle where \( \sin x = \frac{\sqrt{3}}{2} \). The sine of an angle is equal to \frac{\sqrt{3}}{2} when the angle is \( \frac{\pi}{3} \) or \( \frac{2\pi}{3} \).
02

Generalize the Solutions

The sine function is periodic with a period of \( 2\pi \). So, the general solutions are \( x = \frac{\pi}{3} + 2k\pi \) and \( x = \frac{2\pi}{3} + 2k\pi \), where \( k \) is any integer.
03

Write the General Solution

Combine the results from the previous step: The solutions are \( x = \frac{\pi}{3} + 2k\pi \) or \( x = \frac{2\pi}{3} + 2k\pi \).

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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 vital concept in trigonometry. Imagine a circle with a radius of 1 unit, centered at the origin of a coordinate plane.
The unit circle helps us understand the relationships between angles and the coordinates of points on a circle. For any angle \( \theta \), you can find a point \((x, y)\) on the unit circle where \( x \) is the cosine of \( \theta \), and \( y \) is the sine of \( \theta \).
This means if we plot the angle \( \frac{\pi}{3} \), for instance, the sine value corresponds to the \( y \)-coordinate of the point on the unit circle.
Sine Function
The sine function is one of the basic trigonometric functions and is crucial in solving trigonometric equations.
It takes an angle and returns the y-coordinate of the corresponding point on the unit circle. For example, \( \sin(\frac{\u03c0}{3}) = \frac{\u221a3}{2} \).
Key sine function properties to remember include:
  • It ranges between -1 and 1.
  • The sine of \( 0 \) is 0.
  • The sine of \( \pi \) (or 180 degrees) is also 0.
  • It reaches its maximum value of \( 1 \) at \( \frac{\u03c0}{2} \) and its minimum value of -1 at \( \frac{3\u03c0}{2} \).

Understanding these properties will make it easier to identify angles that satisfy given sine values.
Periodic Functions
Trigonometric functions, like sine, are periodic. This means they repeat their values in regular intervals.
For the sine function, the period is \( 2\u03c0 \) or 360 degrees.
This periodic nature helps us generalize solutions to trigonometric equations.
For instance, in the exercise, we identified that sine \( x \) equals \( \frac{\u221a3}{2} \) at \( \frac{\u03c0}{3} \) and \( \frac{2\u03c0}{3} \). Because of the sine function's periodicity, adding multiples of \( 2\u03c0 \) to these angles still gives a valid solution:
  • \( x = \frac{\u03c0}{3} + 2k\u03c0 \)
  • \( x = \frac{2\u03c0}{3} + 2k\u03c0 \)

Here, \( k \) is any integer. This shows how understanding periodicity is essential for finding all possible solutions.

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Most popular questions from this chapter

Find all real numbers in the interval \([0,2 \pi)\) that satisfy each equation. Round approximate answers to the nearest tenth. $$ 6 \sin ^{2} x-2 \cos x=5 $$

Find the inverse of each function and state the domain and range of \(f^{-1}\) $$f(x)=\sin (2 x) \text { for }-\frac{\pi}{4} \leq x \leq \frac{\pi}{4}$$

Find the approximate value of each expression rounded to two decimal places. $$ \cot ^{-1}(15.6) $$

Find the exact value of each expression without using a calculator or table. a. \(\arcsin (1 / 2)\) b. \(\cos ^{-1}(-1 / 2)\) c. \(\tan ^{-1}(-1)\) d. \(\sin (\pi / 3)\) e. \(\cos (-\pi / 2)\) f. \(\sin ^{-1}(-1)\)

Find an equivalent algebraic expression for each composition. $$ \cos (\arctan (x)) $$
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