Problem 26 - Algebra and Trigonometry

Question

In Exercises \(25-32,\) the unit circle has been divided into eight equal arcs, corresponding to t-values of $$ 0, \frac{\pi}{4}, \frac{\pi}{2}, \frac{3 \pi}{4}, \pi, \frac{5 \pi}{4}, \frac{3 \pi}{2}, \frac{7 \pi}{4}, \text { and } 2 \pi $$ a. Use the \((x, y)\) coordinates in the figure to find the value of the trigonometric function. b. Use periodic properties and your answer from part (a) to find the value of the same trigonometric function at the indicated real number. $$ \begin{aligned} &a.\cos \frac{3 \pi}{4}\\\ &b.\cos \frac{11 \pi}{4} \end{aligned} $$

Step-by-Step Solution

Answer

a) The value of \( \cos(\frac{3\pi}{4}) \) is \( -\frac{1}{\sqrt{2}} \).\nb) The value of \( \cos(\frac{11\pi}{4}) \) is \( -\frac{1}{\sqrt{2}} \).

Step 1: Finding Cosine Value at \( \frac{3\pi}{4} \)

Since \( \frac{3\pi}{4} \) lies in the second quadrant where cosine is negative and we know that \( \cos(\pi - \theta) = -\cos(\theta) \), we can subtract \( \frac{\pi}{4} \) from \( \pi \) to get \( \frac{\pi}{4} \) and then apply the formula: thus \( \cos(\frac{3\pi}{4}) = -\cos(\frac{\pi}{4}) = -\frac{1}{\sqrt{2}} \).

Step 2: Understanding periodic properties of cosine function

The cosine function has a period of \(2\pi\), which means every \(2\pi\), the function repeats its values. Therefore, \( \cos( \theta ) = \cos( \theta + 2\pi \cdot n ) \), where n is an integer.

Step 3: Finding Cosine Value at \( \frac{11\pi}{4} \)

Using the periodic property from step 2, \( \frac{11\pi}{4} = \frac{3\pi}{4} + 2\pi \cdot 2 \), which means \( \cos(\frac{11\pi}{4}) = \cos(\frac{3\pi}{4}) \), so \( \cos(\frac{11\pi}{4}) = -\frac{1}{\sqrt{2}} \).

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical functions related to the angles and ratios in a right triangle. They are foundational in various fields such as physics, engineering, and computer science.

• These functions include sine, cosine, tangent, and their reciprocal functions.
• They are used to describe relationships in triangles and can model periodic phenomena.
• Cosine, specifically, is the ratio of the adjacent side to the hypotenuse in a right triangle when considering an angle.

In the unit circle, a tool often used to study trigonometric functions, the radius is one unit. Consequently, the hypotenuse is always 1, making cosine and sine values directly associated with the x and y coordinates, respectively, of a point on the circle. Understanding these relationships helps us accurately determine values for trigonometric functions at various angles.

Cosine

The cosine (\( ext{cos}\)) of an angle is a specific trigonometric function crucial for understanding wave patterns and oscillations.

• On the unit circle, cosine represents the x-coordinate of a point.
• Particularly, the function of cosine is particularly useful in translating between circular motion and wave behavior.
• In the second quadrant of the unit circle, the cosine values are negative because the x-coordinates are negative.

For example, when finding the cosine of \(rac{3 heta}{4}\), it relies on symmetry in the unit circle. Knowing \( ext{cos}( heta) = ext{cos}(rac{ heta}{4})\) and using symmetry properties, granted us a negative value at \( ext{cos}(rac{3 heta}{4})\). This highlights how understanding cosine in the context of the circle aids intuition.

Periodic Properties

A periodic function is one that repeats values at regular intervals or periods.

• Among these, cosine functions exhibit periodic properties with a period of \(2 heta\).
• This means the function's values repeat every \( 2 heta \).

These periodic properties are essential when calculating angles that exceed one full revolution around the circle or even negative angles.
To find \( ext{cos}(rac{11 heta}{4})\), recognizing that it aligns closely with \( ext{cos}(rac{3 heta}{4})\), thanks to periodicity, is key. Additive properties indicate that as \(2 heta n\) more rotations are added or subtracted, the function conceives similar results. Thus, practitioners see these periodic behaviors as the key to calculating, say, an advanced cosine either in theoretical contexts or practical instances.

Pi Radians

Radians are an alternative measure for describing angles, with pi (\( heta\)) playing a fundamental role.

• The concept of radians aligns more naturally with mathematical calculations involving circles.
• Notably, \( heta\) radians corresponds to 180 degrees.

Using radians, angles can be expressed in terms of multiples of \( heta\), which becomes particularly relevant when referring to the unit circle. It provides a seamless translation of angle measurements into circular arc lengths.
In our problem, \( ext{cos}(rac{3 heta}{4})\) encompasses an angle falling into the second quadrant. Here radians helped in easily associating positions and values, particularly in converting degrees into these consistent measurement terms, which get culled for efficient computing and deeper trigonometric explorations.