Problem 18 - Algebra 2

Question

Find the exact value of each expression. \(\sin 315^{\circ}\)

Step-by-Step Solution

Answer

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

Step 1: Identify the reference angle

The angle given is \(315^{\circ}\). Since it's greater than \(270^{\circ}\) and less than \(360^{\circ}\), it is located in the fourth quadrant. To find the reference angle, we subtract it from \(360^{\circ}\). This gives us: \[ \text{Reference angle} = 360^{\circ} - 315^{\circ} = 45^{\circ}. \]

Step 2: Determine the sign of the sine function in the fourth quadrant

In the fourth quadrant, the sine function is negative. This is because the y-coordinates on the unit circle are negative in this quadrant.

Step 3: Find the value of sin using the reference angle

The reference angle \(45^{\circ}\) is one of the special angles. We know from trigonometric identities that \(\sin 45^{\circ} = \frac{\sqrt{2}}{2}\). Since \(315^{\circ}\) is in the fourth quadrant where sine is negative, we have \(\sin 315^{\circ} = -\frac{\sqrt{2}}{2}\).

Key Concepts

Sine Function

The sine function is one of the fundamental trigonometric functions. It is commonly abbreviated as \( \sin \). The sine of an angle in a right triangle is the ratio of the length of the opposite side to the length of the hypotenuse. This means for an angle \( \theta \), \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \).

When extending to the unit circle, the sine function represents the y-coordinate of a point on the circle at a given angle from the positive x-axis. The unit circle is a circle with a radius of 1 centered at the origin \((0,0)\). This property makes the sine function periodic, with a cycle repeating every \(360^{\circ}\) or \(2\pi\) radians.

• The sine function has a range of values from -1 to 1.
• It is positive in the first and second quadrants, and negative in the third and fourth quadrants.
• The sine function is crucial in modeling wave patterns and in various real-world applications.

Reference Angle

A reference angle is a convenient way to evaluate trigonometric functions for angles greater than \(90^{\circ}\). It is the acute angle, formed by the terminal side of the given angle and the horizontal axis.

To find the reference angle of any given angle, one must use the following steps based on the quadrant in which the angle lies:

• In the first quadrant, the reference angle is the angle itself.
• In the second quadrant, subtract the angle from \(180^{\circ}\).
• In the third quadrant, subtract \(180^{\circ}\) from the angle.
• In the fourth quadrant, subtract the angle from \(360^{\circ}\).

For example, at \(315^{\circ}\), which is in the fourth quadrant, the reference angle is calculated as \(360^{\circ} - 315^{\circ} = 45^{\circ}\). Knowing this makes it easy to apply known values of sine, cosine, and tangent for special angles.

Reference angles are useful because they simplify the process of finding the values of trigonometric functions at non-standard angles, by leveraging the known values from special right triangles.

Unit Circle

The unit circle is a key concept in trigonometry, providing a geometric visualization of trigonometric functions and their properties. It is a circle with a radius of 1 unit, centered at the origin \((0, 0)\) on the coordinate plane. The equation of the unit circle is \(x^2 + y^2 = 1\).

In the unit circle, any angle \(\theta\) is measured from the positive x-axis counterclockwise. The x-coordinate represents the cosine function \(\cos(\theta)\), and the y-coordinate represents the sine function \(\sin(\theta)\).

• Angles in standard position can span from \(0^{\circ}\) to \(360^{\circ}\) or from \(0\) to \(2\pi\) radians.
• The unit circle simplifies the understanding of periodicity and symmetry in trigonometric functions.
• It aids in memorizing values for sine and cosine at common angles like \(30^{\circ}\), \(45^{\circ}\), \(60^{\circ}\), and \(90^{\circ}\).

When understanding \(\sin 315^{\circ}\), one can visualize the angle's terminal side in the fourth quadrant, where sine values are negative. The position on the unit circle helps determine that \(\sin 315^{\circ} = -\frac{\sqrt{2}}{2}\).