Problem 17 - Precalculus

Question

Match the trigonometric function values. a. \(\frac{1}{2}\) b. \(\frac{\sqrt{3}}{2}\) c. \(\frac{\sqrt{2}}{2}\) $$\sin 45^{\circ}$$

Step-by-Step Solution

Answer

The value of \(\sin 45^{\circ}\) is \(\frac{\sqrt{2}}{2}\) which matches option (c).

Step 1: Identify the Trigonometric Value

We need to match the given trigonometric function value for \ \(\sin 45^{\circ}\). Start by identifying what \ \(\sin 45^{\circ}\) equals. By recalling from trigonometry tables or the unit circle, \ \(\sin 45^{\circ} = \frac{\sqrt{2}}{2}\).

Step 2: Match with Given Options

Now, we need to match the identified value of the trigonometric function with the given options: \(\frac{1}{2}\), \(\frac{\sqrt{3}}{2}\), and \(\frac{\sqrt{2}}{2}\). We can see that \(\sin 45^{\circ} = \frac{\sqrt{2}}{2}\) matches option (c), \(\frac{\sqrt{2}}{2}\).

Key Concepts

Understanding the Unit Circle

Understanding the unit circle is crucial in trigonometry. Imagine a standard circle with a radius of 1, centered at the origin of a coordinate plane. This is the unit circle. The circle helps visualize different angles and their respective trigonometric values.

Each point on the unit circle corresponds to an angle, usually measured in radians. The x-coordinate of a point on the circle represents the cosine value, while the y-coordinate represents the sine value.

• The circumference of the entire unit circle is \(2\pi\).
• It helps in determining sine, cosine, and tangent values for different angles.

Using the unit circle makes it easy to see why certain trigonometric values repeat every \(360^{\circ}\) or \(2\pi\) radians. This repetition helps in predicting values for other functions too.

Delving into the Sine Function

The sine function is one of the basic trigonometric functions, crucial for various applications in geometry and physics. It relates a right triangle's opposite side to its hypotenuse.

In the context of the unit circle, the sine of an angle is the y-coordinate of the corresponding point on the circle. For instance:

• At \(45^{\circ}\) or \(\frac{\pi}{4}\) radians, the sine value is \(\frac{\sqrt{2}}{2}\).
• This angle, like others, can be represented on the unit circle, where sine values are repeated at regular intervals.

These properties make sine functions predictable and periodic, which is a highlight when exploring signal processing or wave functions in physics.

Recognizing Trigonometric Values

Identifying trigonometric values is essential for solving various mathematical problems. Trigonometric tables or the unit circle can be valuable resources for this.

Common angles, such as \(0^{\circ}\), \(30^{\circ}\), \(45^{\circ}\), \(60^{\circ}\), and \(90^{\circ}\), have specific values that are often memorized:

• For \(45^{\circ}\), the values are \(rac{\sqrt{2}}{2}\) for both sine and cosine functions.
• Using the unit circle can aid in quickly identifying these values without calculation.

Being able to recognize these allows for matching or predicting more complex trigonometric expressions accurately, thus facilitating deeper mathematical understanding.