Problem 38 - Precalculus

Question

Evaluate each expression, if possible. $$\sin \left(-270^{\circ}\right)+\cos 450^{\circ}$$

Step-by-Step Solution

Answer

The evaluated expression is 1.

Step 1: Convert Angles to Positive

The angles given in the problem are negative and more than a full rotation. We first convert these angles to positive angles within one full rotation (0° to 360°).For \(-270^{\circ}\), adding 360° gives \(-270^{\circ} + 360^{\circ} = 90^{\circ}\).For \(450^{\circ}\), subtracting 360° gives \(450^{\circ} - 360^{\circ} = 90^{\circ}\).So, the problem reduces to \(\sin 90^{\circ} + \cos 90^{\circ}\).

Step 2: Evaluate Trigonometric Values

Evaluate the sine and cosine for the transformed angles.We know:- \(\sin 90^{\circ} = 1\)- \(\cos 90^{\circ} = 0\)Thus, the expression becomes \(1 + 0\).

Step 3: Compute the Expression

Add the values obtained from the trigonometric functions.\(1 + 0 = 1\).This is the final evaluated result of the expression.

Key Concepts

Angle Conversion

When working with angles in trigonometry, it's common to encounter angles that are either negative or exceed 360°. To simplify calculations, we often convert these angles to an equivalent angle within the range of 0° to 360°. This process is known as angle conversion.
For negative angles like \(-270^{\circ}\), we add 360° to find a positive equivalent angle. In this case, \(-270^{\circ} + 360^{\circ} = 90^{\circ}\).
For angles greater than 360°, like 450°, we subtract 360° repeatedly until we get a value between 0° and 360°. Here, \(450^{\circ} - 360^{\circ} = 90^{\circ}\). The goal is to find an equivalent angle that lies within a standard circle rotation, which simplifies further calculations.

Sine and Cosine Values

Trigonometric functions like sine and cosine have specific values at particular angles, which are crucial for solving mathematical problems effectively. Let's discuss the sine and cosine of 90° during our exercise, which we converted from given angles.

• Sine at 90°: The value of \(\sin 90^{\circ}\) is 1. This is because, on the unit circle, the y-coordinate (which represents sine) reaches its maximum value of 1 at this angle.
• Cosine at 90°: Conversely, \(\cos 90^{\circ}\) equals 0. At 90°, the x-coordinate (representing cosine) is 0 on the unit circle.

These values can be memorized as they are common in many trigonometric problems and help with quickly evaluating expressions involving sine and cosine.

Full Rotation

In a circle, a full rotation consists of 360°. Understanding this concept is crucial because it allows us to wrap angles into the familiar 0° to 360° range, making it easier to evaluate trigonometric functions.
Angles that are less than 0° or that exceed 360° can be normalized by either adding or subtracting 360°. This wraps the angle around the circle to an equivalent within the primary 0° to 360° cycle.
In our exercise, we adjusted \(-270^{\circ}\) and 450° to 90° by performing these conversions: The angles were beyond a standard cycle, yet through conversion, they fit neatly into trigonometric evaluations, proving how the concept of a full rotation simplifies calculations in trigonometry.