In this article, we will explore how to evaluate sin x when given the value of cos x 2/9 and the condition that tan x 0. By following a step-by-step approach and using fundamental trigonometric identities, we will determine the exact value of sin x without the need for a calculator.

Step-by-Step Guide to Evaluating sin x

The evaluation of sin x involves several key steps. Let's break down each part of the process:

1. Using the Pythagorean Identity

The Pythagorean identity is a fundamental relationship in trigonometry that states:

sin2 x cos2 x 1

Given that cos x 2/9, we can substitute it into the identity:

sin2 x (2/9)2 1

Simplifying the equation:

sin2 x 4/81 1

Subtract 4/81 from both sides to isolate sin2 x:

sin2 x 1 - 4/81

Convert 1 to a fraction with a denominator of 81:

sin2 x 81/81 - 4/81

Perform the subtraction:

sin2 x 77/81

2. Taking the Square Root

Now, we take the square root of both sides to solve for sin x:

sin x pm; √(77/81)

This simplifies to:

sin x pm; 77/9

3. Determining the Sign of sin x

Given the condition that tan x 0, we need to determine in which quadrant x lies. The tangent function is negative in the second and fourth quadrants. In the second quadrant, sin x 0 and cos x 0. In the fourth quadrant, sin x 0 and cos x 0. Since cos x 2/9 0, x must be in the fourth quadrant.

Therefore, in the fourth quadrant, sin x is negative:

sin x -√(77)/9

Conclusion

Using the Pythagorean identity, we found that sin2 x 77/81, and taking the square root for two possible values. By considering the sign of the tangent function and the cosine value, we determined that sin x is negative in the fourth quadrant. Thus, the final result is:

sin x -√(77)/9

Alternative Method

Here’s another approach using the definition of the tangent function:

The tangent function is defined as:

tan x sin x / cos x

Solve for sin x:

sin x tan x * cos x

Given that tan x 0 and cos x 2/9 0, since the product of a negative and a positive number is negative, sin x must be negative.

Substitute cos x 2/9:

sin x -√(77)/9

Key Takeaways

Related Trigonometric Concepts

This problem showcases the application of fundamental trigonometric principles, such as the Pythagorean identity and the tangent function. Understanding these concepts is crucial for solving more complex trigonometric problems, making it an essential skill for students and professionals in fields such as engineering, physics, and mathematics.