Problem 37 - Precalculus

Question

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

Step-by-Step Solution

Answer

The expression evaluates to 1.

Step 1: Simplify Angles Using Reference Angles

The cosine and sine functions have periodicity of 360°. To simplify these problems, we need to find equivalent angles between 0° and 360°.For \(-270^{\circ}\), add 360°: \(-270^{\circ} + 360^{\circ} = 90^{\circ}\).For \(450^{\circ}\), subtract 360°: \(450^{\circ} - 360^{\circ} = 90^{\circ}\).

Step 2: Evaluate Cosine and Sine Functions at the Simplified Angles

Now that the angles are simplified, we can evaluate them using known values on the unit circle.The cosine of 90° is 0: \(\cos 90^{\circ} = 0\).The sine of 90° is 1: \(\sin 90^{\circ} = 1\).

Step 3: Add the Evaluated Values

The original expression simplifies to \(\cos(-270^{\circ}) + \sin 450^{\circ}\), which we simplified to \(0 + 1\).Thus, the expression evaluates to 1.

Key Concepts

Angle Simplification

When working with trigonometric functions, it is often necessary to simplify angles to make calculations easier. Simplifying angles helps find equivalent angles within a range, typically from 0° to 360°, because trigonometric functions are periodic. This means that they repeat their values in regular intervals.

The angles

• Such as -270° can be simplified by adding 360°, resulting in 90°, and
• 450° can be simplified by subtracting 360°, also resulting in 90°.

This process allows us to evaluate the trigonometric functions without losing any information. The resulting angle, which falls within the primary cycle (0° to 360°), retains the same trigonometric properties as the original angle.

By simplifying angles, calculations become more straightforward, and it's easier to refer to known values on the unit circle.

Unit Circle

The unit circle is an essential tool in trigonometry that helps us understand and evaluate trigonometric functions like sine and cosine. It is a circle with a radius of 1, centered at the origin (0,0) of a coordinate plane. Each angle in the unit circle corresponds to a point

• Where the x-coordinate represents the cosine value, and
• The y-coordinate represents the sine value of that angle.

For instance, at 90° on the unit circle, the coordinates are (0,1). This means that

• The cosine value of 90° is 0, and
• The sine value is 1.

These values remain consistent due to the circle's symmetry and periodic properties of sine and cosine functions. Using the unit circle, we can easily find trigonometric values for any angle by locating its corresponding point on the circle.

Reference Angles

Reference angles are helpful tools for understanding how trigonometric functions behave at different angles. A reference angle is the acute angle formed between the terminal side of the given angle and the x-axis.

To determine the reference angle for any given angle, always consider the absolute difference between the angle and the nearest x-axis, which is usually 0° or 180°. Reference angles provide a way to calculate trigonometric functions for larger angles by using their equivalent acute angle.

In The given problem,

• The reference angle for both -270° and
• 450° is 90°, since they fall in the same quadrant.

This allows us to evaluate trigonometric functions for those angles as if they were simply 90°. This simplification is tremendously helpful for solving trigonometric expressions quickly and accurately.