Problem 22 - Algebra 2

Question

Find the exact value of each trigonometric function. \(\sin 240^{\circ}\)

Step-by-Step Solution

Answer

\(\sin 240^{\circ} = -\frac{\sqrt{3}}{2}\)

Step 1: Determine the Reference Angle

To find the exact value of \(\sin 240^{\circ}\), start by determining the reference angle. Since 240° is in the third quadrant, subtract 180° from 240° to find the reference angle: \(240^{\circ} - 180^{\circ} = 60^{\circ}\). So, the reference angle is \(60^{\circ}\).

Step 2: Apply Sine in the Third Quadrant

In the third quadrant, the sine function is negative because sine is positive in the first and second quadrants, but negative in the third and fourth quadrants. Thus, \(\sin 240^{\circ} = -\sin 60^{\circ}\).

Step 3: Use the Sine of 60 Degrees

From the unit circle or trigonometric ratios, we know \(\sin 60^{\circ} = \frac{\sqrt{3}}{2}\). Since we are in the third quadrant, \(\sin 240^{\circ} = -\frac{\sqrt{3}}{2}\).

Key Concepts

Sine Function

The sine function is one of the primary trigonometric functions, often denoted as \( \sin \theta \). It relates the angle in a right triangle to the ratio of the length of the opposite side to the hypotenuse. In the context of the unit circle, the sine of an angle \( \theta \) corresponds to the y-coordinate of the point where the terminal side of the angle intersects the circle.
This implies that as the angle \( \theta \) changes, the sine value varies between -1 and 1.

• When \( \theta = 0 \) or \( \theta = 180^{\circ} \), \( \sin \theta = 0 \)
• When \( \theta = 90^{\circ} \) or \( \theta = 270^{\circ} \), \( \sin \theta = 1 \) or \( \sin \theta = -1 \) respectively
• The sine function is periodic with a period of \( 360^{\circ} \) or \( 2\pi \) radians

Reference Angle

A reference angle is the smallest angle that a given angle makes with the x-axis. It is always a positive acute angle, usually measured between 0° and 90°. To find the reference angle for any given angle, you need to:

• Identify which quadrant the original angle is in
• Use the reference angle formula based on that quadrant

For an angle in the third quadrant, like 240°, the reference angle \( \theta_{\text{ref}} \) is given by:\[\theta_{\text{ref}} = \theta - 180^{\circ}\]This means for \( 240^{\circ} \), the reference angle is \( 60^{\circ} \). This angle has the same sine value as the original angle, except for the sign, due to the quadrant in which it lies.

Unit Circle

The unit circle is a fundamental concept in trigonometry, representing a circle with a radius of one unit centered at the origin of a coordinate plane. It is an invaluable tool when understanding trigonometric functions as it links angles directly to coordinates on the circle.
Key points to remember about the unit circle:

• Each angle \( \theta \) in the unit circle corresponds to a point \((x, y)\) on the circle, where \( x = \cos \theta \) and \( y = \sin \theta \)
• The entire circle is completed with an angle of \( 360^{\circ} \) or \( 2\pi \) radians
• The sine value of an angle is simply the y-coordinate of the corresponding point on the unit circle

Understanding the unit circle helps you easily find the sine, cosine, and tangent of common angles and understand their periodic properties.

Quadrants

The coordinate plane is divided into four quadrants, which help in determining the sign of a trigonometric function based on the angle's placement:

• Quadrant I: All trigonometric functions are positive
• Quadrant II: Sine and cosecant are positive
• Quadrant III: Tangent and cotangent are positive
• Quadrant IV: Cosine and secant are positive

In the third quadrant, where an angle of \( 240^{\circ} \) lies, both sine and cosine are negative. This explains why \( \sin 240^{\circ} = -\sin 60^{\circ} \). Knowing the quadrant helps in determining the sign of the sine and other trigonometric values efficiently.

Exact Values of Trigonometric Functions

Exact values of trigonometric functions refer to specific, often well-known values for sine, cosine, and tangent at key angles such as 0°, 30°, 45°, 60°, and 90°. These values are often derived from geometric principles or the unit circle.For example, the exact value of \( \sin 60^{\circ} \) is:\[\sin 60^{\circ} = \frac{\sqrt{3}}{2}\]When dealing with angles such as 240°, using the reference angle of 60° and the quadrant's properties, the exact value can easily be determined as \( \sin 240^{\circ} = -\frac{\sqrt{3}}{2} \). Having these exact values at your fingertips is crucial for solving trigonometric problems quickly and accurately.