Problem 6 - Precalculus

Question

Find the exact value of the cosine and sine of the given angle. $$ \theta=\frac{3 \pi}{4} $$

Step-by-Step Solution

Answer

\( \cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2} \) and \( \sin\left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2} \).

Step 1: Identify the Angle in the Unit Circle

The angle given is \( \theta = \frac{3\pi}{4} \). This angle is in the second quadrant of the unit circle. In this quadrant, the reference angle is used to find the sine and cosine values, but note that sine is positive and cosine is negative.

Step 2: Find the Reference Angle

The reference angle for \( \theta = \frac{3\pi}{4} \) can be found by subtracting it from \( \pi \):\[ \pi - \frac{3\pi}{4} = \frac{4\pi}{4} - \frac{3\pi}{4} = \frac{\pi}{4} \].This means the reference angle is \( \frac{\pi}{4} \).

Step 3: Calculate Cosine and Sine for Reference Angle

The sine and cosine of the reference angle \( \frac{\pi}{4} \) are:\[ \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \] \[ \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \].

Step 4: Adjust for Second Quadrant Signs

Since \( \theta = \frac{3\pi}{4} \) is in the second quadrant, the cosine is negative while the sine stays positive:\[ \cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2} \] \[ \sin\left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2} \].

Key Concepts

Understanding the Unit Circle

The unit circle is a fundamental concept in trigonometry. It helps you understand angles and trigonometric functions like sine and cosine. Imagine a circle with a radius of 1, centered at the origin of a coordinate system. This is our unit circle. Every point on the unit circle represents an angle and has coordinates \(x, y\). The x-coordinate represents the cosine of the angle, and the y-coordinate represents the sine of the angle. Here's why the unit circle is so useful:

• It provides a geometric way to visualize angles and their sine and cosine values.
• It helps determine the sign of the trigonometric functions based on the quadrant an angle is in.
• It repeats values in a periodic manner, making calculations easier with known angles like \(\frac{\pi}{4}\) or \(\frac{3\pi}{4}\).

For the angle \(\frac{3\pi}{4}\), it lies in the second quadrant. In this quadrant, sine values are positive but cosine values are negative. Understanding the unit circle allows you to apply these conditions correctly.

What is a Reference Angle?

A reference angle is key to simplifying trigonometric calculations. It is essentially the smallest angle between the terminal side of a given angle and the x-axis. Reference angles make it easier to determine the sine and cosine of angles in different quadrants by relating them to angles in the first quadrant (where both sine and cosine are positive). To find the reference angle for an angle in the second quadrant, you subtract the angle from \(\pi\). For \(\theta = \frac{3\pi}{4}\), the reference angle is \(\pi - \frac{3\pi}{4} = \frac{\pi}{4}\).Key points:

• Reference angles help make trigonometric calculations more straightforward.
• In the second quadrant, you calculate it by subtracting the angle from \(\pi\).
• They effectively utilize known trigonometric values at common reference angles like \(\frac{\pi}{4}\).

Exploring the Cosine Function

The cosine function is one of the fundamental trigonometric functions. It relates the angle in a right-angled triangle to the ratio of the adjacent side over the hypotenuse. On the unit circle, cosine represents the x-coordinate of a point. Cosine properties to remember:

• In quadrant I, both sine and cosine values are positive.
• In quadrant II, sine is positive, and cosine is negative.
• In quadrant III, both sine and cosine values are negative.
• In quadrant IV, cosine is positive, and sine is negative.

For \(\frac{3\pi}{4}\), since it's located in the second quadrant, cosine is negative. Calculate it using the reference angle: \(\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\), but as it is in the second quadrant, the cosine becomes negative: \(\cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2}\).

Understanding the Sine Function

The sine function is another essential trigonometric function. It is the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. On the unit circle, sine represents the y-coordinate. Key points about the sine function:

• In quadrant I, both sine and cosine values are positive.
• In quadrant II, sine is positive, and cosine is negative.
• In quadrant III, both sine and cosine values are negative.
• In quadrant IV, cosine is positive, and sine is negative.

For \(\frac{3\pi}{4}\), since it's located in the second quadrant, sine remains positive. Using the reference angle, its value is \(\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\). The sine for the angle \(\frac{3\pi}{4}\) remains: \(\sin\left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2}\).