Problem 21 - Precalculus with Calculus Previews

Question

For the given value of \(t\) determine the reference angle \(t^{\prime}\) and the exact values of \(\sin t\) and \(\cos t\). Do not use a calculator. $$ t=-\pi / 4 $$

Step-by-Step Solution

Answer

\( t' = \pi/4 \); \( \sin(-\pi/4) = -\frac{\sqrt{2}}{2} \); \( \cos(-\pi/4) = \frac{\sqrt{2}}{2} \).

Step 1: Understand the Reference Angle

The reference angle of an angle \( t \) is the smallest positive angle between the terminal side of \( t \) and the x-axis. It is always positive and less than or equal to \( \pi/2 \). For an angle \( t = -\pi / 4 \), the absolute value is already less than \( \pi/2 \), so the reference angle \( t' = \pi/4 \).

Step 2: Determine the Exact Value of \( \sin \)

Since the original angle \( t = -\pi/4 \) is located in the fourth quadrant, the sine function is negative. The sine of the reference angle \( t' = \pi/4 \) is \( \sin(\pi/4) = \frac{\sqrt{2}}{2} \). Thus, \( \sin(-\pi/4) = -\frac{\sqrt{2}}{2} \).

Step 3: Calculate the Exact Value of \( \cos \)

In the fourth quadrant, the cosine function is positive. The cosine of the reference angle \( t' = \pi/4 \) is \( \cos(\pi/4) = \frac{\sqrt{2}}{2} \). Therefore, \( \cos(-\pi/4) = \frac{\sqrt{2}}{2} \).

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in understanding the relationships between angles and sides in triangles, particularly right triangles. They help describe the properties of waves, circular movement, and various oscillations. The main trigonometric functions are sine, cosine, and tangent. Each of these functions provides unique information about the angle:

• Sine represents the ratio of the length of the opposite side to the hypotenuse.
• Cosine indicates the ratio of the adjacent side to the hypotenuse.
• Tangent shows the ratio between the opposite and the adjacent side.

To visualize these, imagine a unit circle where every point on the circle represents an angle in radians. The x-coordinate describes the cosine value, and the y-coordinate gives the sine value of the angle. Understanding how these trigonometric functions work is essential for a wide range of mathematical problems and real-world applications.

Sine and Cosine Values

Sine and cosine values are particularly important for analyzing angles and their properties. These values often rely on known angles, such as 0, \(\pi/4, \pi/2, \pi\), and others. For the reference angle \(\pi/4\), we have key trigonometric values:

• Sine: \(\sin(\pi/4) = \frac{\sqrt{2}}{2}\)
• Cosine: \(\cos(\pi/4) = \frac{\sqrt{2}}{2}\)

These values are crucial for various calculations in trigonometry. They help in determining the sine and cosine of angles based on their reference angles, especially in non-standard positions. For example, knowing the properties of an angle in the fourth quadrant helps adjust the sine value's sign to negative while keeping the cosine value positive, closely tied to their positions on the Cartesian plane.

Quadrants in Trigonometry

In trigonometry, the x-y plane is divided into four quadrants, crucial for determining the signs of trigonometric functions:

• First Quadrant: Both sine and cosine are positive.
• Second Quadrant: Sine is positive, but cosine is negative.
• Third Quadrant: Both sine and cosine are negative.
• Fourth Quadrant: Sine is negative, while cosine is positive.

When analyzing the angle \(-\pi/4\), it falls within the fourth quadrant. Here, the reference angle \(\pi/4\) is used to retrieve the basic values, but the quadrant determines the sign's alteration. Knowing which quadrant an angle falls into is essential for adjusting the trigonometric values appropriately, as it affects calculations in physics, engineering, and mathematics, making them more accurate and applicable.