Problem 170 - Precalculus

Question

For the following exercises, use a graphing calculator to evaluate. $$ \cos \left(\frac{5 \pi}{6}\right) \cos \left(\frac{2 \pi}{3}\right) $$

Step-by-Step Solution

Answer

The product is \(\frac{\sqrt{3}}{4}\).

Step 1: Identify Angles in Degrees

Convert the angles \(\frac{5\pi}{6}\) and \(\frac{2\pi}{3}\) from radians to degrees. The conversion formula is \(\text{Degrees} = \text{Radians} \times \frac{180}{\pi}\). This gives:\[\frac{5\pi}{6} \times \frac{180}{\pi} = 150^\circ\]\[\frac{2\pi}{3} \times \frac{180}{\pi} = 120^\circ\]

Step 2: Find Cosine Values

Use a graphing calculator or trigonometric table to find the cosine of the angles identified in Step 1. For \(150^\circ\), \(\cos(150^\circ) = -\frac{\sqrt{3}}{2}\). For \(120^\circ\), \(\cos(120^\circ) = -\frac{1}{2}\).

Step 3: Multiply Cosine Values

Multiply the cosine values obtained in Step 2.\[-\frac{\sqrt{3}}{2} \times -\frac{1}{2} = \frac{\sqrt{3}}{4}\]

Key Concepts

Trigonometric Functions

Trigonometric functions are a cornerstone of mathematics, especially when dealing with angles and triangles. These functions relate the angles of a triangle to the lengths of its sides. In trigonometry, the primary functions are sine, cosine, and tangent, each of which forms the basis for various trigonometric identities and equations.

• Sine (\(\sin\)): This function calculates the ratio of the opposite side to the hypotenuse in a right-angled triangle.
• Cosine (\(\cos\)): Cosine gives the ratio of the adjacent side over the hypotenuse.
• Tangent (\(\tan\)): Tangent is the ratio of the opposite side to the adjacent side.

These functions are cyclic and exhibit periodic behavior, meaning they repeat values at regular intervals. They are crucial in modeling waves, harmonic oscillations, and various periodic phenomena in nature.
To further understand the significance of trigonometric functions, remember that they expand beyond triangles, becoming functions of any angle—integral in the study of circles and oscillatory motion.

Cosine Function

The cosine function is one of the most important trigonometric functions, often denoted as \(\cos\). It represents the x-coordinate of a point on the unit circle as it relates to a given angle.

• Range: The values for cosine lie between -1 and 1.
• Periodic Nature: Cosine is periodic with a cycle that repeats every \(360^\circ\) or \(2\pi\) radians.
• Wave Representation: Cosine forms a wave-like graph that peaks at \(1\) and troughs at \(-1\).
• Quadrantal Angles: At \(0^\circ\), \(90^\circ\), \(180^\circ\), \(270^\circ\), and \(360^\circ\)—cosine takes values of \(1\), \(0\), \(-1\), \(0\), and \(1\), respectively.

To evaluate the cosine of any angle, one can use unit circle properties or trigonometric identities. Cosine values for standard angles, like \(30^\circ\), \(45^\circ\), and \(60^\circ\), often show up in various applications, making this function a vital part of geometrical calculations.

Angle Conversion

Angle conversion is an essential concept when working with trigonometric functions, especially in different measurement systems. The two most common units for measuring angles are degrees and radians.

• Degree: A circle is commonly divided into \(360\) degrees. It's a familiar unit used in daily life and simple geometric calculations.
• Radian: A radian relates an angle to the arc length and is based on the radius of a circle. One complete revolution around a circle is \(2\pi\) radians.

Converting between these units is crucial for consistent calculations. The conversion factor is based on the circle's total \(360^\circ\) equaling \(2\pi\) radians, thus:

• To convert from radians to degrees, multiply by \(\frac{180}{\pi}\).
• To convert from degrees to radians, multiply by \(\frac{\pi}{180}\).

By understanding these conversions, you can seamlessly switch between angle measures, making calculations more intuitive and applicable across various math and physics problems.

Radians to Degrees

Understanding how to convert radians to degrees is necessary for interpreting or solving many trigonometric problems, as radians are often used in calculus and scientific contexts, whereas degrees are more intuitive for everyday situations.
The basic formula to convert an angle from radians to degrees is: \[ \text{Degrees} = \text{Radians} \times \frac{180}{\pi} \]Let's apply this formula to convert \(\frac{5\pi}{6}\) and \(\frac{2\pi}{3}\) radians, as seen in the exercise:

• \(\frac{5\pi}{6}\) radians: \(\frac{5\pi}{6} \times \frac{180}{\pi} = 150^\circ\)
• \(\frac{2\pi}{3}\) radians: \(\frac{2\pi}{3} \times \frac{180}{\pi} = 120^\circ\)

These conversions make it easier to interpret and work with trigonometric functions, as degrees often provide a more practical context for understanding angles in geometric settings and in correlation with standard trigonometric values.