Problem 73
Question
Use the negative-angle identities to compute the exact value of each of the given trigonometric functions. $$\sin \left(-\frac{2 \pi}{3}\right)$$
Step-by-Step Solution
Verified Answer
The exact value of \( \sin \left(-\frac{2 \pi}{3}\right) \) is \( -\frac{\sqrt{3}}{2} \)
1Step 1: Write down the given function
The given function is \( \sin \left(-\frac{2 \pi}{3}\right) \)
2Step 2: Apply the negative-angle identity
Using the negative-angle identity, the given function can be rewritten as \( -\sin \left(\frac{2 \pi}{3}\right) \)
3Step 3: Calculate the value of sine function
Now calculate the value of \( \sin \left(\frac{2 \pi}{3}\right) \). Looking at the unit circle, or using common trigonometric values, we find \( \sin \left(\frac{2 \pi}{3}\right) = \frac{\sqrt{3}}{2} \)
4Step 4: Apply the calculated value
Substituting this value into the equation from Step 2 gives \( -\frac{\sqrt{3}}{2} \)
Key Concepts
Trigonometric FunctionsExact Values of Trigonometric FunctionsUnit Circle
Trigonometric Functions
Trigonometric functions are fundamental in the study of mathematics, particularly in fields like geometry, calculus, and physics. These functions relate the angles of a triangle to the lengths of its sides and are applicable to both right-angled and non-right-angled triangles. The primary trigonometric functions are sine (sin), cosine (cos), and tangent (tan), each providing a ratio between sides of a right-angled triangle. The cosecant (csc), secant (sec), and cotangent (cot) are their respective reciprocals.
Understanding these functions thoroughly is key to solving complex trigonometric problems. For example, recognizing that the sine function represents the ratio of the length of the side opposite the angle to the hypotenuse of a right triangle will aid in visualizing and memorizing important trigonometric values. These functions are also periodic, meaning they repeat their values in regular intervals, as seen in their respective graphs, which illustrate their behavior over the span of angle measures.
Understanding these functions thoroughly is key to solving complex trigonometric problems. For example, recognizing that the sine function represents the ratio of the length of the side opposite the angle to the hypotenuse of a right triangle will aid in visualizing and memorizing important trigonometric values. These functions are also periodic, meaning they repeat their values in regular intervals, as seen in their respective graphs, which illustrate their behavior over the span of angle measures.
Exact Values of Trigonometric Functions
While calculator approximations can be useful, knowing the exact values of trigonometric functions is vital for precise calculations, particularly in mathematics and engineering. The exact values are based on certain angles for which the trigonometric functions can be calculated without approximation.
These specific angles, typically multiples of 30° (or \(\pi/6\) radians), 45° (or \(\pi/4\) radians), and 60° (or \(\pi/3\) radians), produce trigonometric values that are often square roots of integers over 2. For example, the sine of 60° or \(\pi/3\) is \(\sqrt{3}/2\), an exact value. It's important to become familiar with the exact values for the unit circle angles as they frequently appear in both coursework and standardized tests. Additionally, they serve as the foundation for using and deriving various trigonometric identities—including the negative-angle identities highlighted in the given exercise—facilitating the understanding of more complicated trigonometric expressions.
These specific angles, typically multiples of 30° (or \(\pi/6\) radians), 45° (or \(\pi/4\) radians), and 60° (or \(\pi/3\) radians), produce trigonometric values that are often square roots of integers over 2. For example, the sine of 60° or \(\pi/3\) is \(\sqrt{3}/2\), an exact value. It's important to become familiar with the exact values for the unit circle angles as they frequently appear in both coursework and standardized tests. Additionally, they serve as the foundation for using and deriving various trigonometric identities—including the negative-angle identities highlighted in the given exercise—facilitating the understanding of more complicated trigonometric expressions.
Unit Circle
The unit circle is a critical concept in trigonometry, defined as a circle with a radius of one unit, centered at the origin of a coordinate plane. It offers an insightful way to visualize and understand trigonometric functions and their values for all angle measures. Points on the unit circle can be described using coordinates \( (x, y) \), which correspond to \( (\cos(\theta), \sin(\theta)) \) where \( \theta \) is the angle formed by the line segment from the origin to the point and the positive x-axis.
The unit circle also helps to make sense of the periodic nature of trigonometric functions and allows the determination of exact values for sine and cosine at various angles. For example, to find the exact value of \( \sin(-2\pi/3) \) in the original exercise, one would locate the corresponding point on the unit circle in the negative angle direction and then use the y-coordinate, considering the negative-angle identity \( \sin(-\theta) = -\sin(\theta) \). By referencing or visualizing the unit circle, one quickly reaffirms that for \( 2\pi/3 \) radians, the y-coordinate (the sine value) is positive \( \sqrt{3}/2 \), therefore \( \sin(-2\pi/3) \) would be \( -\sqrt{3}/2 \), demonstrating the utility of the unit circle in finding exact values of trigonometric functions.
The unit circle also helps to make sense of the periodic nature of trigonometric functions and allows the determination of exact values for sine and cosine at various angles. For example, to find the exact value of \( \sin(-2\pi/3) \) in the original exercise, one would locate the corresponding point on the unit circle in the negative angle direction and then use the y-coordinate, considering the negative-angle identity \( \sin(-\theta) = -\sin(\theta) \). By referencing or visualizing the unit circle, one quickly reaffirms that for \( 2\pi/3 \) radians, the y-coordinate (the sine value) is positive \( \sqrt{3}/2 \), therefore \( \sin(-2\pi/3) \) would be \( -\sqrt{3}/2 \), demonstrating the utility of the unit circle in finding exact values of trigonometric functions.
Other exercises in this chapter
Problem 72
In this set of exercises, you will use right triangle trigonometry to study real-world problems. Unless otherwise indicated, round answers to four decimal place
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The horizontal range of a projectile that is fired with an initial velocity \(v_{0}\) at an acute angle \(\theta\) with respect to the horizontal is given by $$
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Convert the angle measures given in DMS form to decimal degrees with three decimal places. $$120^{\circ} 50^{\prime} 15^{\prime \prime}$$
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Using appropriate graphs, show that \(\cos (-x)=\cos x\)
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