Problem 86 Suppose $\theta$ is not an int... [FREE SOLUTION]

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Precalculus A Prelude to Calculus

Sheldon Axler

$Math Studyset 91影视 Explanations$ Math

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Chapter 6: Problem 86

Suppose $\theta$ is not an integer multiple of $\pi .$ Explain why the point $(1,2 \cos \theta)$ is on the line containing the point $(\sin \theta, \sin (2 \theta))$ and the origin.

Short Answer

Expert verified

To show that the point $(1, 2\cos\theta)$ lies on the line containing the origin and the point $(\sin \theta, \sin(2\theta))$, we calculate the slopes of both lines passing through the origin and the given points, and show that they are equal. The slope of the line passing through the origin and $(\sin \theta, \sin(2\theta))$ is $\frac{\sin(2\theta)}{\sin \theta}$, while the slope of the line passing through the origin and $(1, 2\cos\theta)$ is $2\cos\theta$. Using the trigonometric identity $\sin(2\theta) = 2\sin\theta\cos\theta$, we find that both slopes are equal, meaning that all three points lie on the same line as long as $\theta$ is not an integer multiple of $\pi$.

Step by step solution

Calculate the slope of the line passing through the origin and point $(\sin \theta, \sin(2\theta))$

To find the slope of a line, we can use the formula: \[slope = \frac{y2-y1}{x2-x1}\] In this case, the coordinates of the two points are $(0, 0)$ (origin) and $(\sin \theta, \sin(2\theta))$. Thus, the slope of this line can be found as follows: \[slope = \frac{\sin(2\theta) - 0}{\sin \theta - 0} = \frac{\sin(2\theta)}{\sin \theta}\]

Calculate the slope of the line passing through the origin and point $(1, 2\cos\theta)$

Similarly, the coordinates of the two points are $(0, 0)$ (origin) and $(1, 2\cos\theta)$. Thus, the slope of this line can be found as follows: \[slope = \frac{2\cos\theta - 0}{1 - 0} = 2\cos\theta\]

Show that both slopes are equal

Now we need to show that both slopes calculated above are equal. Using trigonometric identity of the double angle for sine, we have: \[\sin(2\theta) = 2\sin\theta\cos\theta\] Now, divide both sides by $\sin\theta$: \[\frac{\sin(2\theta)}{\sin\theta} = 2\cos\theta\] Notice that both slopes found above are equal, which means that the three points $(0,0)$, $(\sin \theta, \sin(2\theta))$, and $(1, 2\cos\theta)$ lie on the same line. Thus, we've proven that the point $(1, 2\cos\theta)$ is on the line containing the point $(\sin \theta, \sin(2\theta))$ and the origin, as long as $\theta$ is not an integer multiple of $\pi$.

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91影视

Suppose \(\theta\) is not an integer multiple of \(\pi .\) Explain why the point \((1,2 \cos \theta)\) is on the line containing the point \((\sin \theta, \sin (2 \theta))\) and the origin.

Short Answer

Step by step solution

Calculate the slope of the line passing through the origin and point \((\sin \theta, \sin(2\theta))\)

Calculate the slope of the line passing through the origin and point \((1, 2\cos\theta)\)

Show that both slopes are equal

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