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if cosecant is equal to 9/2 then find contangent

if cosecant is equal to 9/2 then find contangent

2 min read 21-01-2025
if cosecant is equal to 9/2 then find contangent

Meta Description: Learn how to find the cotangent of an angle when the cosecant is known. This guide provides a clear, step-by-step solution using trigonometric identities and right-angled triangle properties, perfect for students and math enthusiasts.

This article will guide you through calculating the cotangent of an angle given that its cosecant is 9/2. We'll leverage trigonometric identities and the properties of right-angled triangles to reach the solution.

Understanding the Relationship Between Cosecant and Cotangent

Before diving into the calculation, let's refresh our understanding of cosecant and cotangent.

  • Cosecant (csc): The cosecant of an angle is the reciprocal of the sine of that angle. In other words, csc(θ) = 1/sin(θ).

  • Cotangent (cot): The cotangent of an angle is the reciprocal of the tangent of that angle. Or, cot(θ) = 1/tan(θ) = cos(θ)/sin(θ).

We are given that csc(θ) = 9/2. This means sin(θ) = 1/csc(θ) = 2/9.

Using Trigonometric Identities to Find Cotangent

We can use the Pythagorean identity to relate sine and cosine: sin²(θ) + cos²(θ) = 1.

Since we know sin(θ) = 2/9, we can substitute this into the Pythagorean identity to solve for cos(θ):

(2/9)² + cos²(θ) = 1

cos²(θ) = 1 - (2/9)² = 1 - 4/81 = 77/81

cos(θ) = ±√(77/81) = ±√77/9

Now that we have both sin(θ) and cos(θ), we can calculate cot(θ):

cot(θ) = cos(θ)/sin(θ) = (±√77/9) / (2/9) = ±√77/2

Visualizing with a Right-Angled Triangle

Alternatively, consider a right-angled triangle. Since csc(θ) = hypotenuse/opposite = 9/2, we can label the hypotenuse as 9 and the opposite side as 2.

Using the Pythagorean theorem (a² + b² = c²), we can find the adjacent side:

adjacent² + 2² = 9² adjacent² = 81 - 4 = 77 adjacent = √77

Now, we can directly calculate the cotangent:

cot(θ) = adjacent/opposite = √77/2

The Ambiguity of the Sign

Notice that we obtained both positive and negative values for cot(θ). This ambiguity arises because the cosecant function is positive in both the first and second quadrants. Without knowing the specific quadrant of angle θ, we cannot definitively determine whether the cotangent is positive or negative.

Therefore, the final answer is:

cot(θ) = ±√77/2

Conclusion

We successfully determined the cotangent given the cosecant using trigonometric identities and geometrical interpretation. Remember to consider the quadrant of the angle to resolve the sign ambiguity in the cotangent value. This problem highlights the interconnectedness of trigonometric functions and the importance of understanding their relationships. Understanding this concept is crucial for more advanced trigonometry and calculus problems.

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