Problem 42 - Precalculus with Calculus Previews

Question

Find the given trigonometric function value. Do not use a calculator. $$ \sin 270^{\circ} $$

Step-by-Step Solution

Answer

\(\sin 270^{\circ} = -1\).

Step 1: Understanding the Quadrant

The angle 270° lies on the y-axis between the third and fourth quadrants. Specifically, it is at the intersection of the negative y-axis.

Step 2: Trigonometric Value on the Axis

The sine of any angle is the y-coordinate of the corresponding point on the unit circle. For 270°, we are directly on the negative y-axis.

Step 3: Analyzing the Unit Circle

On the unit circle, at 270°, the coordinates of the point are (0, -1). Hence, \(\sin 270^{\circ} = -1\).

Step 4: Final Solution

Therefore, the trigonometric value of \(\sin 270^{\circ}\) is -1.

Key Concepts

Understanding the Unit Circle

The unit circle is a fundamental concept in trigonometry and helps in understanding the behavior of trigonometric functions. Imagine a circle with a radius of 1 unit, centered at the origin of a coordinate plane. This is what we call the unit circle. Every point on this circle can be described by coordinates \((x, y)\), which are equivalent to \((\cos \theta, \sin \theta)\) for an angle \(\theta\) measured from the positive x-axis.

The unit circle allows us to easily understand and visualize angles. These angles are usually measured in degrees or radians. Each point on the circle correlates to a unique angle, allowing us to find trigonometric values, like sine and cosine, without extensive calculations. Also, since the radius is always 1, the circle helps maintain consistent relationships among angles and their trigonometric values.

This concept is crucial because it simplifies the calculation of trigonometric functions for key angles, such as \(90^{\circ}, \ 180^{\circ}, \ 270^{\circ}, \) and \(360^{\circ}\). These angles lie exactly on the axes, making it straightforward to determine their sine and cosine values.

Exploring the Sine Function

The sine function is one of the primary trigonometric functions and plays a crucial role in analyzing periodic phenomena, such as sound waves or light. For any angle \(\theta\) on the unit circle, the sine function is defined as the y-coordinate of the corresponding point. This means that for every angle, \(\sin \theta\) represents how far up or down the point is from the center of the circle.

• For \(\theta = 0^{\circ}\), the sine value is 0.
• At \(\theta = 90^{\circ}\), the sine reaches its maximum value of 1, as the point is at its highest on the unit circle.
• Moving to \(\theta = 180^{\circ}\), the sine drops back to 0, as the point reaches the x-axis again.
• At \(\theta = 270^{\circ}\), which is our point of interest, the sine value becomes -1, the lowest point on the y-axis.

Understanding these key angles and their sine values helps in visualizing how the sine function behaves throughout the entire circle, which is critical when dealing with trigonometric problems.

Navigating Through the Quadrants

When dealing with trigonometry, it's essential to recognize which quadrant an angle lies in, as this influences the sign and sometimes the value of trigonometric functions. The Cartesian coordinate plane, divided into four quadrants, serves as a reference frame:

• Quadrant I: Angles from \(0^{\circ}\) to \(90^{\circ}\). Here, both x and y coordinates are positive, meaning all trigonometric functions are positive.
• Quadrant II: Angles from \(90^{\circ}\) to \(180^{\circ}\). In this region, the x-coordinate is negative, and the y-coordinate is positive, making sine positive while cosine and tangent are negative.
• Quadrant III: Angles from \(180^{\circ}\) to \(270^{\circ}\). Both coordinates are negative here, resulting in sine and cosine being negative, while tangent is positive.
• Quadrant IV: Angles from \(270^{\circ}\) to \(360^{\circ}\). The x-coordinate is positive, and the y-coordinate is negative, making cosine positive and sine and tangent negative.

The angle \(270^{\circ}\) rests directly on the negative y-axis. It is technically between the third and fourth quadrants, often considered on its axis rather than inside a quadrant. Thus, it does not conform to the typical sign rules of the quadrants, which gives \(\sin 270^{\circ} = -1\). Knowing these characteristics of quadrants makes analyzing angles and their trigonometric functions intuitive and accessible.