Problem 47 - Algebra 2

Question

Find the exact value of each expression. \(\sin \left(-135^{\circ}\right)\)

Step-by-Step Solution

Answer

\(\sin(-135^\circ) = -\frac{\sqrt{2}}{2}\).

Step 1: Determine the Reference Angle

To find the exact value of \( \sin(-135^\circ) \), we first determine its reference angle. Since -135° is in the fourth quadrant (moving clockwise from 0°), we add 360° to find the equivalent positive angle: \(-135^\circ + 360^\circ = 225^\circ\). The reference angle is then \(225^\circ - 180^\circ = 45^\circ\).

Step 2: Use the Unit Circle

From the unit circle, we know the sine value of the reference angle \(45^{\circ}\). \(\sin(45^\circ) = \frac{\sqrt{2}}{2}\). Since -135° corresponds to 45° in the fourth quadrant, where sine values are negative, use \(\sin(-135^\circ) = -\sin(45^\circ)\).

Step 3: Add the Negative Sign

Apply the conclusion from Step 2: the sine is negative in the fourth quadrant. Therefore, \(\sin(-135^\circ) = -\frac{\sqrt{2}}{2}\).

Key Concepts

Reference Angle

A reference angle is a helpful tool in trigonometry that allows us to simplify the computation of trigonometric functions for any angle. It is the acute angle formed by the terminal side of the angle and the horizontal axis. The reference angle is always between 0° and 90°. To find the reference angle for any given angle, we consider the angle's position within the four quadrants of the coordinate system. When dealing with a negative angle such as -135°, it indicates that we are moving clockwise from the positive x-axis. To convert a negative angle into a positive one and find its reference angle, we can add 360° to make it positive. For -135°, this results in a positive angle of 225°. However, this is not the final reference angle. We have to look at how far this angle is from the x-axis, which leads us to the reference angle being: - 225° - 180° = 45°.

Unit Circle

The unit circle is a vital concept in trigonometry. It is a circle with a radius of one unit centered at the origin of the coordinate plane. This simple geometric shape offers a comprehensive way to understand the behavior of trigonometric functions. As we move around the unit circle, each angle corresponds to a point on the circumference where the x-coordinate represents the cosine of the angle, and the y-coordinate represents the sine. For instance, at 45°, which is a common reference angle, the coordinates are \(rac{ ext{√}2}{2}, rac{ ext{√}2}{2}\). Understanding the unit circle allows us to determine sine and cosine values quickly, as they are simply the coordinates on the unit circle. When dealing with an angle beyond 90°, or with negative angles like -135°, we recognize it as equivalent to 225° on the unit circle. Importantly, the sine value at 225° or -135° maintains the same magnitude but must be considered for its sign based on quadrants.

Quadrants

The coordinate plane is divided into four sections called quadrants, which help us determine the sign of trigonometric functions. These quadrants start from the positive x-axis and are numbered counterclockwise:

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

When dealing with trigonometric functions like sine, it's essential to consider which quadrant the terminal side of the angle lies in, as it affects the sign of the output. For an angle like -135°, after converting it to 225°, you will find it's located in Quadrant III. The sine value for Quadrant III is negative, which is why\(-135°\)'s sine is negative \\(-rac{\sqrt{2}}{2}\). Understanding which quadrant an angle lies in helps identify why certain trigonometric functions assume their specific signs.