Calculate Trigonometric Functions Step-by-Step Guide

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In the realm of trigonometry, understanding the relationships between angles and their trigonometric functions is paramount. When provided with the value of one trigonometric function and the quadrant in which the angle lies, we can systematically determine the values of the remaining trigonometric functions. This article delves into the process of calculating the values of the remaining trigonometric functions of an angle alpha, given that tan(alpha) = -1/3 and 90° < alpha < 180°. This exercise will not only demonstrate the application of fundamental trigonometric identities but also highlight the significance of quadrant analysis in determining the signs of trigonometric functions.

The problem at hand involves determining the values of the remaining trigonometric functions of an angle alpha, given that tan(alpha) = -1/3 and 90° < alpha < 180°. This information tells us two crucial things: the tangent of the angle is negative, and the angle lies in the second quadrant. Recall that the trigonometric functions are defined as ratios of sides in a right triangle, and their signs vary depending on the quadrant in which the angle lies. In the second quadrant, sine is positive, while cosine and tangent are negative. This knowledge is essential for correctly determining the signs of the remaining trigonometric functions.

1. Understanding the Given Information

The prompt is to find the other trigonometric functions where tan(α) = -1/3 and 90° < α < 180°. This places α in the second quadrant, where sine is positive, and cosine and tangent are negative. We will utilize this information to accurately determine the signs of the remaining trigonometric functions.

2. Using the Tangent Identity

Tangent, being the ratio of sine to cosine (tan(α) = sin(α) / cos(α)), provides our first clue. Given tan(α) = -1/3, we understand the ratio between sine and cosine, but we need to determine their individual values. The negative sign confirms our understanding that either sine or cosine is negative, but not both, within the specified quadrant.

3. Applying the Pythagorean Identity

The Pythagorean identity, sin²(α) + cos²(α) = 1, is our next tool. We need to express sine or cosine in terms of the other using the tangent value. From tan(α) = -1/3, we can infer a relationship between sine and cosine, but first, let’s manipulate the Pythagorean identity to incorporate the tangent.

To relate tangent to the Pythagorean identity, we first recall that tan(α) = sin(α) / cos(α). From this, we can establish a relationship between sin(α) and cos(α). Since tan(α) = -1/3, we can write sin(α) = -1/3 * cos(α) or cos(α) = -3 * sin(α). However, we need to be cautious about the signs here. We know that 90° < α < 180°, which means alpha lies in the second quadrant, where sine is positive and cosine is negative. Therefore, we can express sin(α) in terms of cos(α) as sin(α) = -(1/3)cos(α), or cos(α) in terms of sin(α) as cos(α) = -3sin(α). The negative sign is crucial here to reflect the correct quadrant.

Now, we substitute this relationship into the Pythagorean identity, sin²(α) + cos²(α) = 1. We can substitute either sin(α) or cos(α), but let's substitute cos(α) = -3sin(α) into the identity:

sin²(α) + (-3sin(α))² = 1

This simplifies to:

sin²(α) + 9sin²(α) = 1

Combining like terms, we get:

10sin²(α) = 1

Now, we can solve for sin²(α):

sin²(α) = 1/10

Taking the square root of both sides, we get:

sin(α) = ±√(1/10)

Since alpha lies in the second quadrant, where sine is positive, we choose the positive root:

sin(α) = √(1/10) = √10 / 10

Thus, we have found the value of sin(α).

4. Calculating Cosine

Now that we know sin(α) = √10 / 10, we can calculate cos(α) using the relationship we established earlier: cos(α) = -3sin(α). Substituting the value of sin(α), we get:

cos(α) = -3(√10 / 10) = -3√10 / 10

This confirms that cosine is negative in the second quadrant, as expected.

5. Determining the Remaining Trigonometric Functions

With sine, cosine, and tangent determined, we can easily find the remaining functions:

  • Cosecant (csc α): The reciprocal of sine. csc(α) = 1 / sin(α) = 1 / (√10 / 10) = 10 / √10 = √10
  • Secant (sec α): The reciprocal of cosine. sec(α) = 1 / cos(α) = 1 / (-3√10 / 10) = -10 / (3√10) = -√10 / 3
  • Cotangent (cot α): The reciprocal of tangent. cot(α) = 1 / tan(α) = 1 / (-1/3) = -3

6. Summary of Results

In summary, given tan(α) = -1/3 and 90° < α < 180°, the values of the trigonometric functions are:

  • sin(α) = √10 / 10
  • cos(α) = -3√10 / 10
  • tan(α) = -1/3 (Given)
  • csc(α) = √10
  • sec(α) = -√10 / 3
  • cot(α) = -3

The quadrant analysis was a critical step in this problem. By knowing that 90° < α < 180°, we could confidently determine the signs of sine, cosine, and other functions. This exemplifies the importance of considering the quadrant when working with trigonometric functions, as it directly affects the sign of the function value. Without this analysis, we might end up with incorrect signs for some of the trigonometric functions.

The ability to calculate trigonometric functions is not just an academic exercise; it has practical applications in various fields, including:

  • Navigation: Determining the position and direction of a vessel or aircraft.
  • Engineering: Designing structures, bridges, and other infrastructure.
  • Physics: Analyzing wave phenomena, oscillations, and other physical systems.
  • Computer graphics: Creating realistic 3D models and animations.

Determining the values of trigonometric functions given one function value and the quadrant requires a sound understanding of trigonometric identities and the significance of quadrant analysis. By systematically applying the Pythagorean identity and considering the signs of functions in different quadrants, we can accurately calculate the values of the remaining trigonometric functions. This problem demonstrates the power of trigonometric principles and their applicability in various real-world scenarios. By understanding these concepts, one can tackle complex problems in mathematics, physics, engineering, and other scientific disciplines. The trigonometric functions are fundamental tools that empower us to analyze and understand the world around us, from the simple oscillations of a pendulum to the intricate movements of celestial bodies.

For those eager to delve deeper into trigonometry, several avenues for further exploration exist:

  1. Trigonometric Identities: Investigating a wider array of trigonometric identities can unlock more efficient problem-solving techniques. Identities such as the sum and difference formulas, double-angle formulas, and half-angle formulas can significantly simplify complex trigonometric expressions and equations.

  2. Inverse Trigonometric Functions: Understanding inverse trigonometric functions (arcsin, arccos, arctan) allows you to find angles corresponding to specific trigonometric ratios. This knowledge is crucial for solving trigonometric equations and finding angles in various applications.

  3. Applications in Physics and Engineering: Exploring how trigonometric functions are used in physics and engineering can provide valuable insights into their practical significance. For example, understanding simple harmonic motion, wave propagation, and vector analysis requires a solid grasp of trigonometry.

  4. Complex Numbers and Trigonometry: The connection between complex numbers and trigonometric functions is a fascinating area to explore. Euler's formula, which relates complex exponentials to trigonometric functions, has profound implications in mathematics and physics.

By engaging with these topics, you can deepen your understanding of trigonometry and its role in various fields of study. The world of trigonometry is vast and interconnected, offering endless opportunities for discovery and application.