Introduction

This article will guide you through the process of determining the values of cosine and tangent (cos A and tan A) when given the sine of an angle (sin A 3/4). We will explore various methods, including the Pythagorean identity and definitions, and discuss how these values depend on the quadrant. This information is crucial for understanding trigonometric identities and their applications in various fields such as engineering, physics, and mathematics.

Challenging the Given Information

A common mistake when solving for cos A and tan A is to assume cos A 5/4. However, this solution is incorrect and underestimates the complexity involved. Let's delve into the correct approach step by step.

Step 1: Using the Pythagorean Identity

The Pythagorean identity, a fundamental property of trigonometric functions, states that:

sin^2 A cos^2 A 1

Given that sin A 3/4, we can substitute this value into the identity to solve for cos A.

Substitution and Isolation

We start by squaring the sine value:

sin^2 A (3/4) ^ 2 9/16

Substituting this into the identity:

9/16 cos^2 A 1

Isolating cos^2 A:

cos^2 A 1 - 9/16 7/16

Taking the square root to find cos A:

cos A ±√(7/16) ±sqrt{7}/4

Step 2: Calculating Tangent (tan A)

The tangent of an angle is defined as the ratio of the sine to the cosine:

tan A sin A / cos A

Substituting the known values:

tan A (3/4) / (±sqrt{7}/4) 3/±sqrt{7}

To rationalize the denominator:

tan A ±(3sqrt{7})/7

Summary of Results

The values of cos A and tan A are:

cos A ±sqrt{7}/4 tan A ±(3sqrt{7})/7

The signs depend on the quadrant in which angle A is located. For the principal solutions in the range 0 ≤ A ≤ 2π:

The acute angle (first quadrant): cos A sqrt{7}/4, tan A 3/sqrt{7} The obtuse angle (second quadrant): cos A -sqrt{7}/4, tan A -3/sqrt{7}

Understanding the Significance

The variation in signs of cos A and tan A based on the quadrant is crucial. This knowledge is essential for solving trigonometric equations in the context of real-world applications where the angle's location in the coordinate plane can significantly impact the solution.

Conclusion

By leveraging the Pythagorean identity and the definitions of cosine and tangent, we can accurately determine the values of these trigonometric functions. Understanding how different quadrants affect these values is vital for a comprehensive grasp of trigonometry. Whether you're in a classroom setting, working on a project, or pursuing a career in a STEM field, the ability to solve for cos A and tan A when given sin A is an invaluable skill.

Keywords

sin A, cos A, tan A, Pythagorean identity, trigonometric functions