Introduction to Trigonometry

Trigonometry is a branch of mathematics that deals with relationships involving lengths and angles of triangles. It is widely used in various fields such as physics, engineering, and navigation. This article aims to explain a specific problem related to trigonometric functions, focusing on the value of tan(90° - x) given that csc(x) 3 where x is an acute angle.

Understanding the Problem Statement

The problem requires finding the value of tan(90° - x) given that csc(x) 3. The first step involves understanding the relationship between different trigonometric functions and how they can be manipulated to find the desired value.

Solution Using Trigonometric Identities

To solve this problem, we will use the cofunction identity and basic trigonometric relationships. The cofunction identity states that tan(90° - x) cot(x).

Step 1: Identify the Cofunction

From the problem, we know that:

csc(x) 3

Using the co-function identity:

tan(90° - x) cot(x)

Step 2: Find the Sine and Cosine of x

Given that csc(x) 3, we can find the sine of x:

sin(x) 1/csc(x) 1/3

Using the Pythagorean identity, we find the cosine of x:

sin2(x) cos2(x) 1

(1/3)2 cos2(x) 1

1/9 cos2(x) 1

cos2(x) 1 - 1/9 8/9

cos(x) √(8/9) 2√2/3

Since x is an acute angle, cos(x) is positive.

Step 3: Calculate the Tangent of x

Using the definition of tangent:

tan(x) sin(x) / cos(x) (1/3) / (2√2/3) 1 / 2√2 √2 / 4

Step 4: Find the Cotangent of x

The cotangent of x is the reciprocal of the tangent:

cot(x) 1 / tan(x) 4 / √2 2√2

Final Answer

Thus, the value of tan(90° - x) is:

boxed{2√2}

Alternative Method Using a Right Triangle

This problem can also be solved using a right triangle approach:

Step 1: Construct the Right Triangle

Given that csc(x) 3, we know:

sin(x) 1/3

This means that in a right triangle with hypotenuse 3 and opposite side 1, the adjacent side can be found using the Pythagorean theorem:

adjacent2 opposite2 hypotenuse2

adjacent2 12 32

adjacent2 9 - 1 8

adjacent √8 2√2

Step 2: Find the Tangent of (90° - x)

The tangent of (90° - x) is the ratio of the opposite to the adjacent side in the right triangle:

tan(90° - x) opposite / adjacent 2√2 / 1

boxed{2√2}

Conclusion

To summarize, we demonstrated two methods for finding the value of tan(90° - x) given that csc(x) 3. The first method uses trigonometric identities, while the second uses a right triangle approach. Both methods lead to the same result: tan(90° - x) 2√2.