Which Expression Is Equivalent To Sin 7pi 6

Which Expression is Equivalent to sin(7π/6)? A Comprehensive Guide

Determining the equivalent expression for sin(7π/6) involves understanding the unit circle, trigonometric identities, and the properties of sine functions. This comprehensive guide will explore various approaches to solve this problem, providing a detailed explanation for each method. We'll also delve into related concepts to build a stronger foundation in trigonometry.

Understanding the Unit Circle

The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. It's a fundamental tool for understanding trigonometric functions. Each point on the unit circle can be represented by its coordinates (cos θ, sin θ), where θ is the angle formed between the positive x-axis and the line segment connecting the origin to that point.

Locating 7π/6 on the Unit Circle

The angle 7π/6 radians is equivalent to 210 degrees. This angle lies in the third quadrant of the unit circle. In the third quadrant, both the x-coordinate (cosine) and the y-coordinate (sine) are negative. This crucial piece of information will guide our calculations.

Method 1: Using the Unit Circle Directly

The most straightforward method involves directly reading the sine value from the unit circle at the angle 7π/6. As mentioned earlier, the sine value in the third quadrant is negative. The reference angle for 7π/6 is π/6 (or 30 degrees). The sine of π/6 is 1/2. Therefore, since we are in the third quadrant, the sine of 7π/6 is -1/2.

Method 2: Using the Angle Sum/Difference Identities

We can express 7π/6 as the sum or difference of known angles. For instance, we can write 7π/6 as π + π/6. This allows us to use the angle sum identity for sine:

sin(A + B) = sin A cos B + cos A sin B

In our case, A = π and B = π/6. Substituting these values:

sin(7π/6) = sin(π + π/6) = sin π cos(π/6) + cos π sin(π/6)

Since sin π = 0 and cos π = -1, the equation simplifies to:

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

Method 3: Utilizing the Periodicity of Sine

The sine function is periodic with a period of 2π. This means that sin(x + 2πk) = sin(x) for any integer k. We can express 7π/6 as:

7π/6 = π/6 + 2πk (where k is an integer)

However, this approach doesn't directly simplify the calculation. Instead, let's consider the relationship between sin(x) and sin(x + π). We know that:

sin(x + π) = -sin(x)

Therefore, we can express 7π/6 as π/6 + π:

sin(7π/6) = sin(π/6 + π) = -sin(π/6) = -1/2

Method 4: Employing the Reference Angle

The reference angle is the acute angle formed between the terminal side of the angle and the x-axis. For 7π/6, the reference angle is π/6. Since 7π/6 lies in the third quadrant where sine is negative, the sine of 7π/6 is the negative of the sine of its reference angle:

sin(7π/6) = -sin(π/6) = -1/2

Understanding the Significance of the Quadrant

The quadrant in which the angle lies is crucial in determining the sign of the trigonometric function. Remember the acronym ASTC (All Students Take Calculus):

• I Quadrant (0 to π/2): All trigonometric functions are positive.
• II Quadrant (π/2 to π): Only sine and cosecant are positive.
• III Quadrant (π to 3π/2): Only tangent and cotangent are positive.
• IV Quadrant (3π/2 to 2π): Only cosine and secant are positive.

Since 7π/6 lies in the third quadrant, only tangent and cotangent are positive; therefore, sine (and cosine) must be negative.

Expanding on Trigonometric Identities

Several trigonometric identities can be used to manipulate and simplify trigonometric expressions. Here are a few relevant ones:

• sin²x + cos²x = 1 (Pythagorean Identity)
• sin(A + B) = sin A cos B + cos A sin B (Angle Sum Identity)
• sin(A - B) = sin A cos B - cos A sin B (Angle Difference Identity)
• sin(2x) = 2sin x cos x (Double Angle Identity)
• sin(-x) = -sin(x) (Odd Function Property)

Understanding and applying these identities is essential for solving more complex trigonometric problems.

Practical Applications of Sine Function

The sine function has widespread applications in various fields, including:

• Physics: Describing oscillatory motion (like simple harmonic motion of a pendulum), wave phenomena (sound waves, light waves), and projectile motion.
• Engineering: Analyzing alternating current circuits, calculating forces and stresses in structures, and designing signal processing systems.
• Computer Graphics: Generating smooth curves and representing periodic phenomena.
• Navigation: Calculating distances and positions using triangulation methods.

Mastering the sine function is crucial for anyone working in these fields.

Conclusion: sin(7π/6) = -1/2

Through various methods—direct use of the unit circle, application of trigonometric identities, consideration of periodicity and reference angles—we've conclusively shown that the expression equivalent to sin(7π/6) is -1/2. Understanding these different approaches solidifies your grasp of trigonometric concepts and equips you to tackle more challenging problems. Remember to always consider the quadrant of the angle to determine the correct sign of the trigonometric function. The more you practice, the more intuitive these calculations will become. Continue exploring trigonometric identities and their applications to broaden your understanding and problem-solving skills.