SOLVED:Finding The Terminal Point For \pi / 3 Now That You Know The ...

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Finding the Terminal Point for $\pi / 3$ Now that you know the terminal point determined by $t=\pi / 6$ , use symmetry to find the terminal point determined by $t=\pi / 3$ (see the figure). Explain your reasoning. Finding the Terminal Point for $\pi / 3$ Now that you know the terminal point determined by $t=\pi / 6$ , use symmetry to find the terminal point determined by $t=\pi / 3$ (see the figure). Explain your reasoning. Algebra and Trigonometry James Stewart,… 2nd Edition Chapter 7, Problem 56 View All Chapters

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Step 1: First, we recall the terminal point for $\pi / 6$ on the unit circle, which is $(\sqrt{3}/2, 1/2)$. Show more…

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Key Concepts

Unit Circle The unit circle is a circle with a radius of one centered at the origin of the coordinate plane. It serves as a foundational concept in trigonometry since each angle in standard position corresponds to a point on the circle, with the x-coordinate representing the cosine of the angle and the y-coordinate representing the sine of the angle. Terminal Point The terminal point of an angle is the point on the unit circle where the terminal side of that angle intersects the circle. This point is crucial as its coordinates (cosine and sine of the angle) define the trigonometric values associated with the angle. Symmetry in the Unit Circle Symmetry in the unit circle refers to the fact that the circle is symmetric about the x-axis, y-axis, and the origin. This property allows one to determine the coordinates of terminal points for related angles by reflecting a known point across these axes. For example, if the terminal point of one angle is known, symmetry can be used to find the terminal point for another angle that is a mirror image with respect to an axis. Reference Angle A reference angle is the acute angle formed by the terminal side of an angle and the horizontal axis. It simplifies the process of finding trigonometric values for angles outside the first quadrant because these values can be derived from the corresponding acute angle, with appropriate sign adjustments based on the quadrant in which the original angle lies.

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DISCOVER = PROVE: Finding the Terminal Point for $\pi / 3$ Now that you know the terminal point determined by $t=\pi / 6,$ use symmetry to find the terminal point determined by $t=\pi / 3$ (see the figure). Explain your reasoning.

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Terminal Points Find the terminal point $P(x, y)$ on the unit circle determined by the given value of $t .$ $$ t=-3 \pi $$

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Terminal Points Find the terminal point $P(x, y)$ on the unit circle determined by the given value of $t .$ $$ t=\frac{3 \pi}{2} $$

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Transcript

00:01 So in this problem, we are told that we know the terminal point determined by t equals pi over 6. 00:05 Well, first off, let's recall what that is. 00:07 Remember, pi over 6 is equivalent to a 30 -degree angle. 00:11 So if we're thinking about our terminal point, that would mean that our x value is 1 half, or sorry, root 3 over 2, switching it here, and that the y value is 1 half. 00:24 So now we want to use this information in order to find the terminal point for t equals pi over 3. 00:31 Well, let's think about where this point would go in terms of our coordinate plane. 00:35 Well, as we mentioned before, pi over 6 is a 30 -degree angle. 00:38 So that would be something like this. 00:40 So this is 30 degrees. 00:41 That gives us this terminal point on our unit circle, root 3 over 2, comma 1 half... Need help? Use Ace Ace is your personal tutor. It breaks down any question with clear steps so you can learn. Start Using Ace Ace is your personal tutor for learning Step-by-step explanations Instant summaries Summarize YouTube videos Understand textbook images or PDFs Study tools like quizzes and flashcards Listen to your notes as a podcast

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