Problem 31
Question
Find the number(s) \(x\) in the interval \([0.2 \pi]\) which satisfy the equation. $$\sin x=1 / 2$$.
Step-by-Step Solution
Verified Answer
The values of \(x\) that satisfy the equation \(\sin x = \frac{1}{2}\) over the interval \([0,2\pi]\) are \(x = \frac{\pi}{6}\) and \(x = \frac{5\pi}{6}\)
1Step 1: Understanding Sinusoidal Values
The sin function has a range of -1 to 1, and it is periodic over the interval \([0, 2\pi]\). It takes the values for \(\sin(0) = 0\), \(\sin(\frac{\pi}{2}) = 1\), \(\sin(\pi) = 0\), \(\sin(\frac{3\pi}{2}) = -1\), and \(\sin(2\pi) = 0\) in the interval given.
2Step 2: Identifying Relevant Angles
The value \(\sin x = \frac{1}{2}\) happens for \(x = \frac{\pi}{6}\) and \(x = \frac{5\pi}{6}\). Check your knowledge about sin values if these values aren't familiar to you.
3Step 3: Verifying the Solution
We know that \(\sin x = \frac{1}{2}\) for \(x = \frac{\pi}{6}\) and \(x = \frac{5\pi}{6}\). From the given interval \([0,2\pi]\), both these solutions hold true. Therefore, the solutions of the equation \(\sin x = \frac{1}{2}\) over the interval \([0,2\pi]\) are \(x = \frac{\pi}{6}\) and \(x = \frac{5\pi}{6}\)
Key Concepts
Sine FunctionPeriodicity of Trigonometric FunctionsUnit Circle
Sine Function
The sine function is an essential part of trigonometry. It is used to determine the ratio of the opposite side to the hypotenuse in a right-angled triangle. This trigonometric function is often written as \( \sin(x) \).
Its values range from -1 to 1, making it a bounded function. This means no matter the angle, the sine will never exceed this range.
A full sine wave completes over the interval \([0, 2\pi]\), starting from 0 and reaching its maximum of 1 at \(\frac{\pi}{2}\). Then it decreases to 0 at \(\pi\), reaches -1 at \(\frac{3\pi}{2}\), and finally completes at \(2\pi\) going back to 0. Understanding these points helps visualize how the sine wave behaves over one complete cycle.
Its values range from -1 to 1, making it a bounded function. This means no matter the angle, the sine will never exceed this range.
A full sine wave completes over the interval \([0, 2\pi]\), starting from 0 and reaching its maximum of 1 at \(\frac{\pi}{2}\). Then it decreases to 0 at \(\pi\), reaches -1 at \(\frac{3\pi}{2}\), and finally completes at \(2\pi\) going back to 0. Understanding these points helps visualize how the sine wave behaves over one complete cycle.
Periodicity of Trigonometric Functions
Trigonometric functions like sine are periodic. This means they repeat their values in regular intervals. For sine, this periodicity is \(2\pi\). In simpler terms, every \(2\pi\) units along the x-axis, the sine function graph will start repeating its pattern.
Understanding periodicity helps in solving trigonometric equations, like finding the angle \(x\) where \(\sin(x) = \frac{1}{2}\). This is because if you find one solution, additional solutions can be found just by adding or subtracting \(2\pi\). However, within a restricted interval, like \([0, 2\pi]\), you only consider the solutions that fall within this range.
Understanding periodicity helps in solving trigonometric equations, like finding the angle \(x\) where \(\sin(x) = \frac{1}{2}\). This is because if you find one solution, additional solutions can be found just by adding or subtracting \(2\pi\). However, within a restricted interval, like \([0, 2\pi]\), you only consider the solutions that fall within this range.
- Basic Period: \(2\pi\) for sine and cosine
- Relates to oscillation and waves
- Helps navigate angle solutions on the unit circle
Unit Circle
The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of 1, centered at the origin of a coordinate plane. The unit circle connects angles and trigonometric functions by using the coordinates of points on the circle to represent the values of sine and cosine.
The x-coordinate of any point on the unit circle gives the cosine of the angle, while the y-coordinate gives the sine of the angle. For instance, when you encounter the equation \(\sin(x) = \frac{1}{2}\), you can locate the corresponding angle by finding where the y-coordinate equals \(\frac{1}{2}\).
The specific angles which meet this condition are \(\frac{\pi}{6}\) and \(\frac{5\pi}{6}\) on the unit circle. These angles correspond to two points where the sine function reaches \(\frac{1}{2}\).
The x-coordinate of any point on the unit circle gives the cosine of the angle, while the y-coordinate gives the sine of the angle. For instance, when you encounter the equation \(\sin(x) = \frac{1}{2}\), you can locate the corresponding angle by finding where the y-coordinate equals \(\frac{1}{2}\).
The specific angles which meet this condition are \(\frac{\pi}{6}\) and \(\frac{5\pi}{6}\) on the unit circle. These angles correspond to two points where the sine function reaches \(\frac{1}{2}\).
- Visual link between angle measures and sine/cosine values
- Helps in solving trigonometric equations
- Facilitates converting between radians and degrees
Other exercises in this chapter
Problem 31
Form the composition \(f \circ g \circ h\) and give the domain. $$f(x)=4 x , \quad g(x)=x-1 , \quad h(x)=x^{2}$$
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Give the domain of the function and sketch the graph. f(x)=1
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Indicate on a number line the numbers \(x\) that satisfy the condition. \(|x-4| \leq 2\).
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Solve the inequality and express the solution set as an interval or as the union of intervals. $$0
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