Problem 16 - Contemporary Precalculus

Question

In Exercises \(15-29,\) find the exact value of the sine, cosine, and tangent of the number, without using a calculator. $$2 \pi / 3$$

Step-by-Step Solution

Answer

Question: Determine the sine, cosine, and tangent of the angle \(2\pi / 3\) without using a calculator. Answer: $$\sin{\frac{2\pi}{3}} = \frac{\sqrt{3}}{2}, \ \cos{\frac{2\pi}{3}} = -\frac{1}{2}, \ \tan{\frac{2\pi}{3}} = -\sqrt{3}$$

Step 1: Convert radians to degrees

To get a better understanding of the angle's position, convert \(2\pi / 3\) radians to degrees. To convert radians to degrees, use the conversion factor: $$1 \ radian = \frac{180}{\pi} \ degrees$$ So, $$2\pi / 3 \equiv \frac{2\pi}{3} \cdot \frac{180}{\pi}$$ $$2\pi / 3 = 120°$$

Step 2: Identify the reference angle and special triangle

Since 120° is in the second quadrant and is a special angle, it lies within the 30-60-90 triangle family. The reference angle in this case would be: $$\angle{A} = 180° - 120° = 60°$$ This means that our special triangle is a 30-60-90 triangle with a 60° reference angle in the second quadrant.

Step 3: Identify the sine, cosine, and tangent of the reference angle

Find the sine, cosine and tangent of the reference angle (60°) using the properties of a 30-60-90 triangle. In a 30-60-90 triangle: $$\sin{A} = \frac{\sqrt{3}}{2}$$ $$\cos{A} = \frac{1}{2}$$ $$\tan{A} = \sqrt{3}$$ Here, we used A to represent 60° to simplify the presentation.

Step 4: Determine the sine, cosine, and tangent of the given angle

Now, we need to determine the sine, cosine, and tangent of 120° using the reference angle values. In the second quadrant, the sine is positive, the cosine is negative, and the tangent is negative. So, we have: $$\sin{\frac{2\pi}{3}} = \sin{120°} = \frac{\sqrt{3}}{2}$$ $$\cos{\frac{2\pi}{3}} = \cos{120°} = -\frac{1}{2}$$ $$\tan{\frac{2\pi}{3}} = \tan{120°} = -\sqrt{3}$$ The sine, cosine, and tangent of the angle \(2\pi / 3\) are as follows: $$\sin{\frac{2\pi}{3}} = \frac{\sqrt{3}}{2}, \ \cos{\frac{2\pi}{3}} = -\frac{1}{2}, \ \tan{\frac{2\pi}{3}} = -\sqrt{3}$$

Key Concepts

sine

The sine function is an essential trigonometric function that helps us understand the relationship between the angles and sides of right triangles. In essence, the sine of an angle is the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right triangle. Imagine a right triangle sitting on a flat surface, with one angle labeled as \(A\).

• If the hypotenuse measures as the longest side, the opposite side is perpendicular to the hypotenuse from angle \(A\).
• Sine is then given by \( \sin{A} = \frac{\text{Opposite side}}{\text{Hypotenuse}} \).

Sine values are extensively used in mathematics and physics, especially when working with wave properties and oscillations. For example, at a 60° angle formed by a special 30-60-90 triangle, \( \sin{60°} = \frac{\sqrt{3}}{2} \). This understanding helps to determine the sine of angles like \(120°\), using quadrant rules to refine the sign, since \(120°\) lies in the second quadrant where sine is positive.

cosine

Cosine is another fundamental trigonometric function, closely related to sine. The cosine of an angle in a right triangle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse. When visualizing a right triangle:

• The adjacent side is the side connected to angle \(A\), aside from the hypotenuse.
• The equation for cosine is \( \cos{A} = \frac{\text{Adjacent side}}{\text{Hypotenuse}} \).

Cosine is valuable for calculating waves, rotational motion, and for understanding situations involving symmetry. For a 60° angle, especially in a special triangle setting like 30-60-90, the cosine is \( \frac{1}{2} \). Applying quadrant knowledge, such as the fact that cosine becomes negative in the second quadrant, helps find the cosine value for angles such as 120°, which yields \( \cos{120°} = -\frac{1}{2} \).

tangent

Tangent is another key player in the trigonometric family. Derived from sine and cosine, the tangent of an angle is the ratio of the sine to the cosine of that angle. Simply put:

• The tangent function illustrates \( \tan{A} = \frac{\sin{A}}{\cos{A}} \).
• In a right triangle, it represents the ratio of the opposite side to the adjacent side: \( \tan{A} = \frac{\text{Opposite side}}{\text{Adjacent side}} \).

For a 60° angle, tangent equals \( \sqrt{3} \) in a 30-60-90 triangle. Knowing the quadrant rules, we adjust the sign: angles like 120° fall in the second quadrant where tangent is negative, so \( \tan{120°} = -\sqrt{3} \). The tangent function is critical for analyzing slopes, angles, and various types of curves.

angle conversion

Understanding angle conversion is vital in trigonometry and ensures the accuracy of calculations. Angles can be measured in two main units: degrees and radians.

• Degrees are commonly used and range from 0° to 360°.
  To convert from radians to degrees, you multiply by \( \frac{180}{\pi} \).
• Radians offer a mathematically neat way of measuring angles, with one full rotation equal to \(2\pi\) radians.
• For example, converting \(2\pi/3\) radians to degrees, use \(2\pi/3 \times \frac{180}{\pi} = 120°\).

Conversion is useful, particularly when angles must be calculated or visualized differently, such as in circles or periodic functions, thus bridging intuition with precision.

special triangles

Special triangles provide useful shortcuts for solving trigonometric problems. Particularly, the 30-60-90 and 45-45-90 triangles have unique side-length ratios:

• In a 30-60-90 triangle, the sides are in the ratio 1 : \(\sqrt{3}\) : 2, corresponding to angles 30°, 60°, and 90°, respectively.
• The 45-45-90 triangle, meanwhile, has sides in a ratio of 1 : 1 : \(\sqrt{2}\).

These triangles simplify the process of finding sine, cosine, and tangent values. In the 30-60-90 triangle, knowing one side lets you easily deduce all other sides and the necessary trigonometric functions. Recognizing these patterns aids in quickly solving various trigonometric problems, especially when handling reference angles such as those found in different quadrants.