Problem 61 - Precalculus: A Concise Course

Question

Evaluate the sine, cosine, and tangent of the angle without using a calculator. $$ \frac{5 \pi}{4} $$

Step-by-Step Solution

Answer

The sine, cosine and tangent of the angle \(\frac{5 \pi}{4}\) are \(-\frac{\sqrt{2}}{2}\), \(-\frac{\sqrt{2}}{2}\), and \(1\), respectively.

Step 1: Determine the Quadrant

Determine in which quadrant the angle \(\frac{5 \pi}{4}\) lies. Since \(\pi < \frac{5 \pi}{4} < \frac{3\pi}{2}\), it's in the third quadrant, where both sine and cosine are negative, and tangent is positive.

Step 2: Find the Corresponding Angle in the First Quadrant

Find corresponding angle in the first quadrant. By subtracting \(\pi\) (or \(2\pi\)), we get an equivalent angle in the first quadrant, which is \(\frac{\pi}{4}\). The sine and cosine values for this angle are both \(\frac{\sqrt{2}}{2}\).

Step 3: Consider the Negative Sign

In the third quadrant, both sine and cosine values are negative. Hence, the sine and cosine of our original angle \(\frac{5 \pi}{4}\) are \(-\frac{\sqrt{2}}{2}\).

Step 4: Find the Tangent

Tangent is the ratio of sine to cosine. Therefore the tangent of the angle is \(\frac{-\sqrt{2}/2}{-\sqrt{2}/2} = 1\).

Key Concepts

Sine of an Angle

When working with trigonometric functions, understanding the sine of an angle is essential. The sine function gives us the y-coordinate of a point on the unit circle corresponding to a given angle. In essence, it measures the vertical distance from the x-axis to the point on the circle.

For the angle \(\frac{5\pi}{4}\), we first locate it in the third quadrant, which is between \(\pi\) and \(\frac{3\pi}{2}\). In this quadrant, the sine is negative because the point is below the x-axis.

• The reference angle for \(\frac{5\pi}{4}\) is \(\frac{\pi}{4}\).
• The sine value at \(\frac{\pi}{4}\) in the first quadrant is \(\frac{\sqrt{2}}{2}\).

Considering the sign of the third quadrant, the sine of \(\frac{5\pi}{4}\) is \(-\frac{\sqrt{2}}{2}\).

This negative sign arises because the third quadrant is where both sine and cosine are negative.

Cosine of an Angle

In trigonometry, the cosine function provides the x-coordinate of a point on the unit circle for a specific angle. It essentially measures the horizontal distance from the y-axis to the point on the circle.

For the angle \(\frac{5\pi}{4}\), we once again find ourselves in the third quadrant. Here, the cosine value is negative due to the point being on the left side of the y-axis.

• The reference angle is again \(\frac{\pi}{4}\).
• The cosine value at \(\frac{\pi}{4}\) is \(\frac{\sqrt{2}}{2}\).

Because the cosine is negative in the third quadrant, the cosine of \(\frac{5\pi}{4}\) is \(-\frac{\sqrt{2}}{2}\).

The negative sign reflects the negative horizontal distance.

Tangent of an Angle

The tangent of an angle in trigonometry is the ratio of the sine of the angle to the cosine of the angle. It gives insight into the slope of the line connecting the origin to the point on the unit circle. Understanding the tangent is fundamental when analyzing angles, especially when compared to arithmetic slopes.

For \(\frac{5\pi}{4}\), both sine and cosine values are \(-\frac{\sqrt{2}}{2}\). Calculating the tangent involves dividing the sine by the cosine:

• \(\text{Tangent} = \frac{-\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}}\).

This simplifies to \(1\).

Thus, the tangent of \(\frac{5\pi}{4}\) is positive \(1\), implying a slope of \(1\) for the line in relation to the x-axis. In the third quadrant, the tangent is indeed positive because both divisor and dividend are negative.