Problem 34 - Algebra 2 and Trigonometry

Question

In \(3-44,\) find the exact value. $$ \sin 45^{\circ}+\cos 60^{\circ} $$

Step-by-Step Solution

Answer

The exact value is \(\frac{\sqrt{2} + 1}{2}\).

Step 1: Recall the Unit Circle Values

To solve for \(\sin 45^{\circ}\) and \(\cos 60^{\circ}\), we need to recall the specific values from the unit circle.- \(\sin 45^{\circ}\) is \(\frac{\sqrt{2}}{2}\).- \(\cos 60^{\circ}\) is \(\frac{1}{2}\).

Step 2: Substitute Known Values into the Expression

Replace \(\sin 45^{\circ}\) with \(\frac{\sqrt{2}}{2}\) and \(\cos 60^{\circ}\) with \(\frac{1}{2}\).This gives:\[ \sin 45^{\circ} + \cos 60^{\circ} = \frac{\sqrt{2}}{2} + \frac{1}{2} \]

Step 3: Compute the Sum

To add the fractions \(\frac{\sqrt{2}}{2}\) and \(\frac{1}{2}\), notice that they have the same denominator.When adding fractions with the same denominator, you add the numerators:\[\frac{\sqrt{2}}{2} + \frac{1}{2} = \frac{\sqrt{2} + 1}{2}\]

Step 4: State the Final Result

The sum of \(\sin 45^{\circ} + \cos 60^{\circ}\) is:\[\frac{\sqrt{2} + 1}{2}\]

Key Concepts

Understanding the Unit Circle

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of one, centered at the origin of a coordinate system. The unit circle helps us find the values of trigonometric functions like sine and cosine. These values are based on angles and their corresponding points on the circle.

When you draw a line from the origin to any point on the unit circle, the angle this line makes with the positive x-axis is the angle in question. For instance, an angle of 0 degrees or 0 radians points directly to the right along the x-axis.

• The unit circle has important angles at regular intervals, such as 0, 30, 45, 60, and 90 degrees.
• These angles correspond to specific points on the circle with coordinates representing the cosine and sine values.

This is why the unit circle is a critical tool. It allows us to quickly and easily determine the sine and cosine of key angles.

Exploring Special Angles

Special angles in trigonometry are those that have values that are particularly easy to remember. These angles include 0°, 30°, 45°, 60°, and 90°. They appear often across various trigonometric problems.

Each of these special angles has well-known sine and cosine values.

• For 45°: Both sine and cosine values are \(\frac{\sqrt{2}}{2}\).
• For 60°: The cosine is \(\frac{1}{2}\) and the sine is \(\frac{\sqrt{3}}{2}\).

These angles simplify calculations significantly because their trigonometric values are derived from simple fractions or radicals.

Understanding these angles will give you a strong foundation in solving trigonometric expressions and equations efficiently.

Calculating Trigonometric Values

Trigonometric values are essential for solving geometric problems, especially involving angles and distances. For any given angle on the unit circle, the x-coordinate provides the cosine value, while the y-coordinate offers the sine value.

In practical terms, knowing these values lets you calculate the sine or cosine for typical angles quickly. Let's consider the example of calculating
\(\sin 45^\circ + \cos 60^\circ\):

• From our knowledge of the unit circle, we know that \(\sin 45^\circ\) is \(\frac{\sqrt{2}}{2}\). This corresponds to the y-coordinate at the 45-degree angle on the circle.
• Similarly, \(\cos 60^\circ\) is \(\frac{1}{2}\), matching the x-coordinate at the 60-degree angle.
• When adding these values, both denominators are the same, allowing us to combine the numerators easily, leading to the expression
  \(\frac{\sqrt{2} + 1}{2}\).

Thus, understanding how to determine these trigonometric values from the unit circle is pivotal in mastering many trigonometry problems.