Question

Explain how you would find the value of \(\sin 390^{\circ}\) using periodic properties.

Short answer

\(\sin 390^{\circ} = \frac{1}{2}\)

Step-by-step solution

- Understand the Periodic Property

The sine function is periodic with a period of 360°. This means \[\[\begin{equation} \sin(x) = \sin(x + 360^{\circ}k) \end{equation}\]\] where k is any integer.

- Subtract Full Periods

Subtract full periods of 360° from 390° to find an equivalent angle between 0° and 360°. Therefore, \[\[\begin{equation} 390^{\circ} - 360^{\circ} = 30^{\circ} \end{equation}\]\]

- Find the Sine Value

Using the result from Step 2, \sin 390^{\circ} = \sin 30^{\circ}. The value of \sin 30^{\circ} is \[\frac{1}{2}\text{.}\]

Key concepts

Trigonometric Functions

Trigonometric functions are fundamental in understanding relationships in triangles and waves. They come in six varieties: sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). In this exercise, we focus on the sine function.

The sine function, represented as \(\text{sin} \theta\), gives the y-coordinate of a point on the unit circle corresponding to an angle \(\theta\). It takes any real number angle and maps it to a value between -1 and 1.

When working with trigonometric functions, knowing their behavior over different angles is essential. For example, \(\text{sin} 30^{\text{°}} = \frac{1}{2}\) regardless of how you reach that angle in the unit circle.

Periodic Properties

One key property of trigonometric functions is their periodicity. This means that the function's values repeat at regular intervals. For the sine and cosine functions, this period is \(\text{360° or 2π radians}\).

This periodic nature can be expressed mathematically: \(\text{sin}(x) = \text{sin}(x + 360^{\text{°}}k)\), where \(k\) is any integer. It implies that adding or subtracting full circles (360°) to/from an angle does not change the sine value.

So, to find \(\text{sin} 390^{\text{°}}\), we subtract the full period of 360°: \(\text{390°} - \text{360°} = 30°\). Therefore, \(\text{sin} 390^{\text{°}} = \text{sin} 30^{\text{°}}\). This property simplifies calculations significantly.

Angle Reduction

Angle reduction is the process of converting a given angle into an equivalent one within a specific range, typically between 0° and 360°. This is useful to simplify the calculation of trigonometric values.

Using periodic properties, any angle larger than 360° can be reduced by subtracting multiples of 360° until it falls into the desired range. For example, reducing \(\text{390°}\), we subtract one full period: \(\text{390°} - \text{360°} = 30°\). Equivalent reductions can be made for negative angles by adding multiples of 360°.

For example, reducing \(\text{-150°}\), we add 360°: \(\text{-150°} + \text{360°} = 210°\). Understanding angle reduction helps simplify trigonometric computations and makes it easier to utilize known values like \(\text{sin} 30^{\text{°}}\).