Which Expression Is Equivalent To Sin 7pi 6

Which Expression is Equivalent to sin(7π/6)? A Deep Dive into Trigonometric Identities

Determining the equivalent expression for sin(7π/6) involves understanding the unit circle, trigonometric identities, and the properties of sine function. This seemingly simple question opens doors to a richer understanding of trigonometry, crucial for various fields, including calculus, physics, and engineering. This article provides a comprehensive exploration of this problem, offering multiple approaches and reinforcing key trigonometric concepts.

Understanding the Unit Circle

The unit circle, a circle with a radius of 1 centered at the origin of a coordinate plane, is a fundamental tool in trigonometry. Each point on the unit circle can be represented by coordinates (cos θ, sin θ), where θ is the angle formed between the positive x-axis and the line connecting the origin to the point.

The angle 7π/6 radians lies in the third quadrant of the unit circle. This is because:

• Radians to Degrees: 7π/6 radians * (180°/π) = 210°

This placement in the third quadrant is crucial because it dictates the signs of both the sine and cosine functions in this region. Both sine and cosine are negative in the third quadrant.

Calculating sin(7π/6) Directly

The most straightforward method is using the unit circle and the properties of the sine function. Since 7π/6 is 210°, we can consider its reference angle. The reference angle is the acute angle formed between the terminal side of the angle and the x-axis. In this case, the reference angle is 210° - 180° = 30° or π/6 radians.

Since sin(π/6) = 1/2, and sine is negative in the third quadrant, we have:

sin(7π/6) = -1/2

This is the direct solution, but exploring alternative methods strengthens our understanding.

Utilizing Trigonometric Identities

Several trigonometric identities can help us find equivalent expressions. Let's explore a few:

1. Using the Sine Subtraction Formula

The sine subtraction formula states: sin(A - B) = sin A cos B - cos A sin B

We can express 7π/6 as a difference of two known angles:

7π/6 = π - π/6

Therefore, sin(7π/6) = sin(π - π/6)

Applying the subtraction formula:

sin(π - π/6) = sin π cos(π/6) - cos π sin(π/6)

Since sin π = 0 and cos π = -1, we get:

sin(7π/6) = 0 * cos(π/6) - (-1) * sin(π/6) = sin(π/6) = 1/2

However, remember that sine is negative in the third quadrant, leading us back to our earlier conclusion:

sin(7π/6) = -1/2

2. Using the Sine Addition Formula (Less efficient in this case)

While the addition formula can be used, it's less efficient for this specific problem. It would require expressing 7π/6 as a sum of two angles, leading to a more complex calculation.

3. Using the Co-function Identities (Not directly applicable)

Co-function identities relate sine and cosine of complementary angles (angles that add up to π/2). These are not directly helpful in this case, as 7π/6 is not a complementary angle to any standard angle.

Exploring the Periodicity of the Sine Function

The sine function is periodic, meaning its values repeat every 2π radians (or 360°). This property allows us to find equivalent expressions by adding or subtracting multiples of 2π.

For example:

sin(7π/6 + 2π) = sin(19π/6) = -1/2

sin(7π/6 - 2π) = sin(-5π/6) = -1/2

Visualizing on the Unit Circle

The unit circle offers a clear visual representation. Locating the point corresponding to 7π/6 (210°) reveals that the y-coordinate (which represents the sine value) is negative and equal to -1/2. This visual confirmation reinforces the calculated result.

Connecting to Other Trigonometric Functions

Understanding sin(7π/6) allows us to easily derive the values of other trigonometric functions for this angle:

• cos(7π/6): Cosine is also negative in the third quadrant. The reference angle is π/6, and cos(π/6) = √3/2. Therefore, cos(7π/6) = -√3/2

• tan(7π/6): tan(θ) = sin(θ)/cos(θ). Thus, tan(7π/6) = (-1/2) / (-√3/2) = 1/√3 = √3/3

• csc(7π/6): csc(θ) = 1/sin(θ). Therefore, csc(7π/6) = 1/(-1/2) = -2

• sec(7π/6): sec(θ) = 1/cos(θ). Therefore, sec(7π/6) = 1/(-√3/2) = -2/√3 = -2√3/3

• cot(7π/6): cot(θ) = 1/tan(θ). Therefore, cot(7π/6) = 1/(√3/3) = √3

Applications in Real-World Problems

Understanding trigonometric functions like sin(7π/6) is crucial in various real-world applications:

• Physics: Analyzing oscillatory motion (like pendulums or waves) requires understanding sine and cosine functions.

• Engineering: Designing structures, analyzing circuits, and modeling various systems frequently involve trigonometric calculations.

• Computer Graphics: Creating realistic images and animations relies heavily on trigonometric functions for transformations and rotations.

• Navigation: Determining positions and distances using GPS technology involves trigonometric calculations.

Conclusion: Mastering Trigonometric Equivalencies

Determining which expression is equivalent to sin(7π/6) is more than just a calculation; it's an exercise in understanding the fundamental principles of trigonometry. By utilizing the unit circle, trigonometric identities, and the concept of periodicity, we've confirmed that sin(7π/6) = -1/2. This seemingly simple problem reinforces the importance of mastering these concepts, crucial for success in various scientific and engineering disciplines. The ability to confidently manipulate trigonometric functions forms a solid foundation for more advanced mathematical studies. Remember, consistent practice and visualization are key to solidifying your understanding of trigonometry.