Problem 113 - Precalculus

Question

Use a calculator to evaluate the following expressions. If you get an error, explain why. $$\cos 270^{\circ}$$

Step-by-Step Solution

Answer

\(\cos 270^{\circ} = 0\).

Step 1: Understand the Expression

We are asked to evaluate the trigonometric function \(\cos 270^{\circ}\). The angle is given in degrees and we need to find the value of the cosine of 270 degrees.

Step 2: Recognize the Angle Location

The angle 270° is positioned on the unit circle. It corresponds to the negative y-axis. If you remember the unit circle, 270° points straight down.

Step 3: Apply Unit Circle Knowledge

On the unit circle, the coordinates of any point can be represented as \((\cos \theta, \sin \theta)\). At 270°, the coordinates are \((0, -1)\).

Step 4: Identify the Cosine Value

The cosine of an angle in the unit circle is the x-coordinate of the corresponding point. Hence, \(\cos 270^{\circ} = 0\).

Step 5: Conclude with the Calculator

Using a calculator to find \(\cos 270^{\circ}\) should return 0. If using a calculator, ensure it is set to degrees, not radians.

Key Concepts

Cosine Function

The cosine function is one of the primary trigonometric functions, accompanied by sine and tangent. It's a fundamental part of understanding angles and their properties in a circle. The cosine function, often noted as \( \cos \theta \), relates the angle \( \theta \) to the x-coordinate of a point on the unit circle, a circle with a radius of 1 centered at the origin of a coordinate plane.

When you evaluate \( \cos 270^{\circ} \), you're looking into where the angle 270 degrees falls on this circle. As the cosine function measures the horizontal position, or x-coordinate, of this angle on the unit circle, 270 degrees aligns with the position (0, -1), translating to \( \cos 270^{\circ} = 0 \).

Here’s why: In a full circle, each quadrant corresponds to 90 degrees. The 270-degree angle stretches three-quarters around the circle, ending on the negative y-axis. Since the cosine value at any point is simply the x-value, 270 degrees yields a result of zero. Understanding this helps visualize the directional property and symmetry inherent in trigonometric functions.

Trigonometric Functions

Trigonometric functions are mathematical functions that relate angles to differences in their respective sides in a right-angled triangle. These functions include sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)), among others. They are essential in mathematics, physics, engineering, and various applied sciences for describing oscillatory motions, rotations, and wave properties.

In the context of the unit circle, trigonometric functions can determine the positions of angles and the values of their respective coordinates.

• Sine: This function gives the y-coordinate on the unit circle.
• Cosine: This function provides the x-coordinate, as explored with \( \cos 270^{\circ} = 0 \).
• Tangent: Calculated as sine divided by cosine, this value can describe the angle's slope or steepness.
  
  

These functions assist in converting an angle into a pair of coordinates, showing the relationships between angles and lengths. By using a calculator properly set to the correct mode, you can find these function values for any angle, including those not lying on the primary axes like 270 degrees.

Angle Measurement

Angle measurement can be expressed in degrees or radians, with degrees being the most common in basic trigonometry. Here, the entire circle is divided into 360 degrees, which is a system that most people find intuitive and straightforward.

To find a measurement in degrees, you determine how far around a circle a given point is and describe it numerically between 0 and 360. For example, 0 degrees starts at the positive x-axis and forms a loop back to 360 degrees, returning to the same point. When you reach 270 degrees, you imagine three-fourths of this complete circle, locating the angle at the negative y-axis as covered.

Be mindful that calculators often toggle between degree and radian modes. A common error when evaluating cosine or other trigonometric functions is forgetting to switch to one’s relevant mode. For our exercise, where we evaluated \( \cos 270^{\circ} \), you would verify the calculator is set to degrees, not radians, to avoid inaccurate results. Degrees are intuitive in navigation, rotations, and familiar scenarios, making them practical and accessible for learners.