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Chapter 8: Problem 84

Find the function value using coordinates of points on the unit circle. $$\sin \frac{5 \pi}{6}$$

Short Answer

Expert verified
\(\sin \frac{5 \pi}{6} = \frac{1}{2} \)

Step by step solution

01

Understand the unit circle

The unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane. Important angles in the unit circle are often measured in radians.
02

Determine the reference angle

Calculate the reference angle for \(\frac{5 \pi}{6}\). The reference angle for an angle \(\theta\) is the acute angle formed by the terminal side of \(\theta\) and the x-axis. For \(\frac{5 \pi}{6}\), the reference angle is \(\frac{\pi}{6}\).
03

Determine the coordinates on the unit circle

Find the coordinates of the point where the terminal side of the reference angle \(\frac{\pi}{6}\) intersects the unit circle. The coordinates for \(\frac{\pi}{6}\) are \(\left( \frac{\sqrt{3}}{2}, \frac{1}{2} \right)\).
04

Identify the sign of the sine value

Since \(\frac{5 \pi}{6}\) is in the second quadrant, and in the second quadrant, the sine values are positive, the corresponding sine coordinate for \(\frac{5 \pi}{6}\) is also positive.
05

State the sine value

Therefore, \(\sin \frac{5 \pi}{6}\) has the same sine value as \(\sin \frac{\pi}{6}\), which is \(\frac{1}{2}\).

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Key Concepts

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

Reference Angle
A reference angle helps us determine trigonometric functions in different quadrants. The reference angle is the acute angle formed by the terminal side of the angle and the x-axis. For example, for an angle \( \theta \) like \( \frac{5 \pi}{6} \), we find the reference angle by subtracting \( \pi \) from \( \theta \). This gives us the reference angle \( \frac{\pi}{6} \). Reference angles are always between 0 and \( \frac{\pi}{2} \), making them easier to work with.
Sine Function
The sine function relates the y-coordinate of a point on the unit circle to an angle. For any angle \( \theta \) on the unit circle, \( \sin(\theta) \) equals the y-coordinate of the corresponding point. For instance, with the angle \( \frac{5 \pi}{6} \), we first find the reference angle \( \frac{\pi}{6} \). The sine of \( \frac{\pi}{6} \) is \( \frac{1}{2} \). Since \( \frac{5 \pi}{6} \) is in the second quadrant, the sine value remains positive.
Quadrants
The coordinate plane is divided into four quadrants, each with different sign rules for sine and cosine values.
  • First Quadrant: All coordinates are positive.
  • Second Quadrant: The x-coordinates are negative, but the y-coordinates and sine values remain positive.
  • Third Quadrant: Both x and y-coordinates are negative.
  • Fourth Quadrant: x-coordinates are positive, but y-coordinates and sine values are negative.
For \( \frac{5 \pi}{6} \), located in the second quadrant, the sine value is positive.
Coordinate Plane
The coordinate plane helps us visualize angles and their trigonometric values. It's defined by a horizontal x-axis and a vertical y-axis. The unit circle, centered at the origin \( (0,0) \) with a radius of 1, is essential for understanding trigonometry. Each point on the unit circle represents an angle's sine (y-coordinate) and cosine (x-coordinate). For \( \frac{5 \pi}{6} \), we find the coordinates \( \left( -\frac{\sqrt{3}}{2}, \frac{1}{2} \right) \). The sine value corresponds to the y-coordinate, \( \frac{1}{2} \).

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Most popular questions from this chapter

Fill in the blank with the correct term. Some of the given choices will not be used. $$\begin{array}{ll}\text { angular speed } & \text { cosine } \\ \text { linear speed } & \text { common } \\ \text { acute } & \text { natural } \\\ \text { obtuse } & \text { horizontal line } \\ \text { secant of } \theta & \text { vertical line } \\ \text { cotangent of } \theta & \text { double- angle } \\ \text { identity } & \text { half-angle } \\ \text { inverse } & \text { coterminal } \\ \text { absolute value } & \text { reference angle }\\\ \text { sines }\end{array}$$ In any triangle, the sides are proportional to the ___________________ of the opposite angles.

Classify the function as linear, quadratic, cubic, quartic, rational, exponential, logarithmic, or trigonometric. $$y=\frac{1}{2} x^{2}-2 x+2$$

Graph the equation using a graphing calculator. \(r=\cos 2 \theta-2\) (Peanut)

Classify the function as linear, quadratic, cubic, quartic, rational, exponential, logarithmic, or trigonometric. $$y=\log _{2}(x-2)-\log _{2}(x+3)$$

Express each vector in the form \(a \mathbf{i}+b \mathbf{j}\) and sketch each in the coordinate plane. The unit vectors \(\mathbf{u}=(\cos \theta) \mathbf{i}+(\sin \theta) \mathbf{j}\) for \(\boldsymbol{\theta}=\pi / 6\) and \(\theta=3 \pi / 4 .\) Include the unit circle \(x^{2}+y^{2}=1\) in your sketch.
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