Problem 50 - Algebra 2

Question

Find the exact value of each function. $$ \sin \left(-315^{\circ}\right) $$

Step-by-Step Solution

Answer

\(\sin(-315^{\circ}) = \frac{\sqrt{2}}{2}\).

Step 1: Understanding Negative Angles

Angles can be negative, which means they are measured in the clockwise direction from the positive x-axis. Here, \(-315^{\circ}\) is a negative angle, so we need to add one full circle (\(360^{\circ}\)) to find its equivalent positive angle.

Step 2: Converting to Positive Angle

To convert \(-315^{\circ}\) to a positive angle, add \(360^{\circ}\): \(-315^{\circ} + 360^{\circ} = 45^{\circ}\)Thus, \(-315^{\circ}\) is equivalent to \(45^{\circ}\).

Step 3: Reference Angle and Sine Value

The reference angle for \(45^{\circ}\) is itself, as it is in the first quadrant. Sine for angles in the first quadrant is positive. We know from trigonometric values that:\(\sin 45^{\circ} = \frac{\sqrt{2}}{2}\).

Step 4: Verify the Sine Value Based on Quadrant

Since \(-315^{\circ}\) is equivalent to \(45^{\circ}\) in the first quadrant, the sine value is positive. Thus, \(\sin(-315^{\circ}) = \sin(45^{\circ}) = \frac{\sqrt{2}}{2}\).

Key Concepts

Negative Angles in Trigonometry

Negative angles in trigonometry may seem confusing at first, but they're quite simple to understand. When we measure angles, we often start from the positive x-axis and move in a counterclockwise direction. This results in a positive angle measurement. However, there are instances where we measure in the opposite, clockwise direction. These measurements are what we call negative angles. For example, - A negative angle like \(-315^{\circ}\) means we've moved 315 degrees clockwise from the positive x-axis.To work with negative angles, we can convert them to positive angles by adding multiples of a full circle (\(360^{\circ}\)). This helps us easily find equivalent angles that are easier to handle in trigonometric calculations. Converting \(-315^{\circ}\) involves adding \(360^{\circ}\): \(-315^{\circ} + 360^{\circ} = 45^{\circ}\).So, \(-315^{\circ}\) is equivalent to \(45^{\circ}\), which we'll use in calculations.

Understanding the Sine Function

The sine function is a foundational element of trigonometry, representing a ratio in a right-angled triangle. Specifically, for any given angle, it gives the ratio between the side opposite the angle and the hypotenuse. In the unit circle,- The sine of an angle measures the y-coordinate of a point on the circumference, derived from the central angle measured from the positive x-axis.Some key properties of the sine function include:

• The sine function is periodic with a period of \(360^{\circ}\) or \(2\pi\) radians.
• It has a maximum value of 1 and a minimum value of -1.
• The function is odd, meaning \(\sin(-\theta) = -\sin(\theta)\).

For instance, when considering angles like \(45^{\circ}\), we know that \(\sin(45^{\circ}) = \frac{\sqrt{2}}{2}\), a result arising from the geometry of the special right-angled triangle where two sides are of equal lengths (isosceles right triangle).

The Role of Reference Angles

Reference angles play a crucial role when resolving various trigonometric problems, especially with non-standard angles. A reference angle is defined as the acute angle that a given angle makes with the x-axis. This helps in simplifying trigonometric calculations in any of the four quadrants.Key aspects:- The reference angle always lies between \(0^{\circ}\) and \(90^{\circ}\).- Reference angles are used to find the trigonometric values of angles in different quadrants, as these values only depend on the reference angle itself.For example, the reference angle for \(-315^{\circ}\), calculated from its positive equivalent \(45^{\circ}\), is \(45^{\circ}\). When an angle like \(45^{\circ}\) is in the first quadrant, it serves as its own reference angle. This makes determining the sine or cosine straightforward since we can use known values like \(\sin(45^{\circ}) = \frac{\sqrt{2}}{2}\).

Understanding Quadrants

The coordinate plane is divided into four parts called quadrants, each affecting trigonometric functions differently. These quadrants are numbered counterclockwise starting from the upper right:

• Quadrant I: Both sine and cosine are positive.
• Quadrant II: Sine is positive, but cosine is negative.
• Quadrant III: Both sine and cosine are negative.
• Quadrant IV: Sine is negative, but cosine is positive.

Given \(-315^{\circ}\), when converted to \(45^{\circ}\), it lies in the first quadrant. This is where both sine and cosine have positive values. That's why the calculation confirms \(\sin(-315^{\circ}) = \sin(45^{\circ}) = \frac{\sqrt{2}}{2}\). Knowing about quadrants helps us anticipate the sign (positive or negative) of trigonometric values, which is crucial for accurate problem-solving in trigonometry.