Problem 116 - Precalculus

Question

Use a calculator to evaluate the following expressions. If you get an error, explain why. $$\sin \left(-270^{\circ}\right)$$

Step-by-Step Solution

Answer

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

Step 1: Understanding the Problem

We need to evaluate the sine function for an angle of \( -270^{\circ} \).

Step 2: Understanding Negative Angles

A negative angle means a clockwise rotation. Since \(270^{\circ}\) added to a full circle \(360^{\circ}\) gives \(90^{\circ}\), \(-270^{\circ}\) corresponds to \(-270^{\circ} + 360^{\circ} = 90^{\circ}\).

Step 3: Evaluating the Sine Function

Now that we've determined the equivalent positive angle, we evaluate \( \sin(90^{\circ}) \). From trigonometric values, \( \sin(90^{\circ}) = 1 \).

Key Concepts

Understanding Negative Angles

Angles can be either positive or negative, dictating their direction of rotation. Positive angles rotate counterclockwise, while negative angles rotate clockwise. By convention, when given a negative angle, we often convert it to a positive angle within a full circle to make calculations easier.
For example, a negative angle like \(-270^{\circ}\) can be converted to a positive one by adding \(360^{\circ}\), resulting in \(90^{\circ}\). This conversion helps to maintain the integrity of angle measurements and aids in trigonometric calculations.
Remember, the full circle measurement (360 degrees) is equivalent to one full rotation, and understanding how negative angles relate to their positive counterparts can simplify many problems.

Exploring the Sine Function

The sine function is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of the length of the opposite side to the hypotenuse. It's a periodic function, repeating its values in cycles of \(360^{\circ}\) or \(\frac{2\pi}{3} \).

• Its values range from -1 to 1.
• Sine of \(0^{\circ}\) is 0.
• Sine of \(90^{\circ}\) is 1.
• Sine of \(180^{\circ}\) is 0.
• Sine of \(270^{\circ}\) is -1.
• Sine of \(360^{\circ}\) returns to 0.

These landmarks are crucial for understanding how sine operates through its cycle. When we rotate angles, whether positive or negative, the periodic nature of the sine function allows easy calculation of the function's value.

Angle Conversion

Angle conversion is a helpful tool for simplifying trigonometric problems. Often, angles are measured in degrees or radians, and conversion between these units or from negative to positive angles is necessary.

• A full circle is \(360^{\circ}\) or \(2\pi\) radians.
• To convert degrees to radians, use \( \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \).
• To convert radians to degrees, use \( \text{Degrees} = \text{Radians} \times \frac{180}{\pi} \).

Furthermore, converting a negative angle to a positive one within a \(360^{\circ}\) cycle is often used, as seen in the original problem where \(-270^{\circ}\) becomes \(90^{\circ}\).
Mastering these conversions ensures precision and ease when working with angles in various mathematical problems.