Finding Tan(A) In Quadrant II When Sin(A) Equals 3/4
In trigonometry, we often encounter problems where we are given the value of one trigonometric function and asked to find the values of other trigonometric functions. This exploration delves into a specific instance of this, focusing on determining the value of the tangent function, denoted as tan(A), when the sine function, sin(A), is given as 3/4, and the angle A lies in the second quadrant. To solve this, we will leverage two fundamental trigonometric identities: the Pythagorean identity sin²(A) + cos²(A) = 1 and the quotient identity tan(A) = sin(A) / cos(A). Understanding these identities and their applications is crucial for navigating various trigonometric problems. We will also discuss the significance of the quadrant in determining the sign of trigonometric functions. This comprehensive guide will walk you through the step-by-step process of finding tan(A), ensuring a clear understanding of the underlying principles and calculations involved. By mastering these concepts, you will be well-equipped to tackle similar problems and deepen your understanding of trigonometry.
Problem Statement
Suppose that sin(A) = 3/4. Using the trigonometric identity sin²(A) + cos²(A) = 1 and the trigonometric identity tan(A) = sin(A) / cos(A), find tan(A) in quadrant II. Round the answer to ten-thousandths.
Solution
1. Using the Pythagorean Identity to Find cos(A)
Our first step involves utilizing the Pythagorean identity, which states that the square of the sine of an angle plus the square of the cosine of the same angle equals 1. Mathematically, this is expressed as sin²(A) + cos²(A) = 1. This identity is a cornerstone of trigonometry, derived from the Pythagorean theorem applied to the unit circle. It allows us to relate the sine and cosine of an angle, making it possible to find one if the other is known. In our case, we are given sin(A) = 3/4, so we can substitute this value into the Pythagorean identity to solve for cos(A). This substitution sets the stage for determining the cosine of the angle, which is essential for finding the tangent. Understanding and applying the Pythagorean identity is crucial for solving a wide range of trigonometric problems, as it provides a fundamental relationship between the sine and cosine functions.
We are given that sin(A) = 3/4. We substitute this value into the Pythagorean identity:
(3/4)² + cos²(A) = 1
This equation allows us to isolate cos²(A). By subtracting (3/4)² from both sides, we can determine the value of cos²(A), which is a crucial step in finding cos(A). The process involves basic algebraic manipulation, but it's essential to ensure accuracy in calculations. Squaring the fraction and subtracting it from 1 requires careful attention to detail. This step is not just about finding a numerical value; it's about applying the Pythagorean identity to establish a relationship between sine and cosine. Mastering this manipulation is fundamental for solving various trigonometric problems, as it allows us to move from knowing the sine of an angle to finding its cosine, or vice versa.
9/16 + cos²(A) = 1
Now, we isolate cos²(A):
cos²(A) = 1 - 9/16
To subtract the fractions, we need a common denominator:
cos²(A) = 16/16 - 9/16
cos²(A) = 7/16
Next, we take the square root of both sides to find cos(A). Remember that taking the square root yields both positive and negative solutions:
cos(A) = ±√(7/16)
cos(A) = ±√7 / √16
cos(A) = ±√7 / 4
2. Determining the Sign of cos(A) in Quadrant II
Now that we have both positive and negative values for cos(A), we need to determine which one is correct for our specific problem. This is where the information about the quadrant of angle A becomes crucial. The quadrant in which an angle lies dictates the signs of its trigonometric functions. In the Cartesian coordinate system, each quadrant corresponds to a different combination of positive and negative values for sine, cosine, and tangent. Understanding these sign conventions is essential for accurately solving trigonometric problems. In this step, we'll focus on how the quadrant affects the sign of the cosine function, allowing us to choose the correct value for cos(A).
Since angle A is in quadrant II, we need to consider the sign of cosine in this quadrant. In quadrant II, the x-coordinates are negative, and cosine corresponds to the x-coordinate on the unit circle. Therefore, cosine is negative in quadrant II. This is a fundamental concept in trigonometry, as the quadrant of an angle directly influences the signs of its trigonometric functions. Remembering the sign conventions for each quadrant is essential for solving problems accurately. In this case, knowing that cosine is negative in quadrant II allows us to eliminate the positive solution for cos(A) and focus on the negative one.
Therefore, cos(A) = -√7 / 4
3. Using the Quotient Identity to Find tan(A)
Having found both sin(A) and cos(A), we can now determine tan(A) using the quotient identity. This identity, tan(A) = sin(A) / cos(A), is a fundamental relationship in trigonometry that links the tangent function to the sine and cosine functions. It is derived from the definitions of these functions in terms of the sides of a right triangle or the coordinates on the unit circle. Understanding and applying this identity is crucial for solving various trigonometric problems, as it allows us to find the tangent of an angle when we know its sine and cosine. In this step, we will substitute the known values of sin(A) and cos(A) into the quotient identity to calculate tan(A).
We use the identity:
tan(A) = sin(A) / cos(A)
Substitute the given value of sin(A) and the calculated value of cos(A):
tan(A) = (3/4) / (-√7 / 4)
To divide fractions, we multiply by the reciprocal of the denominator:
tan(A) = (3/4) * (-4 / √7)
Simplify by canceling the 4s:
tan(A) = -3 / √7
4. Rationalizing the Denominator
In mathematics, it is standard practice to rationalize the denominator, which means eliminating any radical expressions from the denominator of a fraction. This is done to simplify the expression and make it easier to work with. In our case, the denominator contains the square root of 7, so we need to multiply both the numerator and denominator by √7. This process ensures that we are not changing the value of the fraction, as we are essentially multiplying by 1. Rationalizing the denominator is a common technique in simplifying expressions involving radicals and is an essential skill in algebra and trigonometry.
To rationalize the denominator, we multiply the numerator and denominator by √7:
tan(A) = (-3 / √7) * (√7 / √7)
tan(A) = -3√7 / 7
5. Approximating tan(A) to Ten-Thousandths
Our final step is to approximate the value of tan(A) to ten-thousandths. This involves using a calculator to find the decimal value of the expression -3√7 / 7. Rounding to ten-thousandths means we need to consider the digit in the fifth decimal place to determine whether to round the fourth digit up or down. This process provides a numerical approximation of tan(A), which is often necessary for practical applications. Understanding how to round numbers to a specific decimal place is a fundamental skill in mathematics and is essential for presenting results in a clear and concise manner. The final result gives us a precise value for tan(A), fulfilling the requirements of the problem.
Now, we approximate the value to ten-thousandths:
tan(A) ≈ -3 * 2.645751311 / 7
tan(A) ≈ -7.937253933 / 7
tan(A) ≈ -1.1339
Final Answer
Therefore, tan(A) ≈ -1.1339 when rounded to ten-thousandths.
In this detailed solution, we successfully found the value of tan(A) given that sin(A) = 3/4 and angle A is in the second quadrant. We achieved this by utilizing the fundamental Pythagorean identity sin²(A) + cos²(A) = 1 and the quotient identity tan(A) = sin(A) / cos(A). The process involved several key steps: first, we used the Pythagorean identity to find cos(A), carefully considering the sign in quadrant II; then, we applied the quotient identity to calculate tan(A); and finally, we rationalized the denominator and approximated the result to ten-thousandths. This problem highlights the interconnectedness of trigonometric functions and the importance of understanding trigonometric identities and quadrant rules. By mastering these concepts, you can confidently solve a wide range of trigonometric problems. The ability to apply these principles is essential for further studies in mathematics, physics, and engineering, where trigonometric functions play a crucial role.
