Find Tan(A) In Quadrant II: A Step-by-Step Guide
Hey guys! Let's dive into a super interesting trig problem today. We're given that sin(A) = 3/4 and we need to find tan(A), but here's the catch: we know that angle A is in Quadrant II. This little piece of information is crucial because it tells us about the signs of our trigonometric functions. Remember, in Quadrant II, sine is positive, cosine is negative, and tangent, being sine divided by cosine, will also be negative. We will explore step by step how to use trigonometric identities to find the value of tan(A) with precision. Grasping these concepts not only helps in solving trigonometric problems but also enhances your broader mathematical toolkit.
Understanding the Basics: Sine, Cosine, and Tangent
Before we jump into the calculations, let's refresh our understanding of the basic trigonometric functions: sine, cosine, and tangent. These functions relate the angles of a right triangle to the ratios of its sides. Specifically:
- Sine (sin): The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse.
- Cosine (cos): The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse.
- Tangent (tan): The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. It can also be expressed as the sine of the angle divided by the cosine of the angle.
These definitions are foundational for understanding and solving trigonometric problems. Additionally, knowing the signs of these functions in different quadrants is key. In the first quadrant (0° to 90°), all trigonometric functions are positive. In the second quadrant (90° to 180°), sine is positive, while cosine and tangent are negative. In the third quadrant (180° to 270°), tangent is positive, and sine and cosine are negative. Finally, in the fourth quadrant (270° to 360°), cosine is positive, while sine and tangent are negative. This knowledge is vital for determining the correct sign of the trigonometric functions in various contexts.
Using the Pythagorean Identity to Find Cosine
Okay, so we know sin(A) = 3/4, and we need to find cos(A) first. This is where the Pythagorean identity comes to our rescue. The Pythagorean identity is a fundamental trigonometric identity that states:
sin²(A) + cos²(A) = 1
This identity is derived from the Pythagorean theorem (a² + b² = c²) applied to the unit circle. It links the sine and cosine of an angle, allowing us to find one if we know the other. Now, let’s plug in the value of sin(A) into this identity:
(3/4)² + cos²(A) = 1
This simplifies to:
9/16 + cos²(A) = 1
To isolate cos²(A), we subtract 9/16 from both sides:
cos²(A) = 1 - 9/16
Which gives us:
cos²(A) = 7/16
Now, to find cos(A), we take the square root of both sides:
cos(A) = ±√(7/16)
So:
cos(A) = ±√7 / 4
But wait! Remember we're in Quadrant II, where cosine is negative. This is a super important detail. Therefore, we choose the negative root:
cos(A) = -√7 / 4
Choosing the correct sign based on the quadrant is a critical step in solving trigonometric equations. Without considering the quadrant, we would end up with two possible values for cosine, only one of which is correct in our context. Therefore, always pay attention to the quadrant information to ensure the accuracy of your solutions.
Calculating Tangent Using sin(A) and cos(A)
Alright, we've got sin(A) = 3/4 and cos(A) = -√7 / 4. Now we can finally calculate tan(A). We know that tangent is defined as:
tan(A) = sin(A) / cos(A)
This identity is one of the most fundamental in trigonometry, linking sine, cosine, and tangent. It's essential for solving various problems, particularly those involving right triangles and the unit circle. Plugging in our values, we get:
tan(A) = (3/4) / (-√7 / 4)
To divide by a fraction, we multiply by its reciprocal:
tan(A) = (3/4) * (-4 / √7)
The 4s cancel out:
tan(A) = -3 / √7
Now, let's rationalize the denominator by multiplying both the numerator and denominator by √7:
tan(A) = (-3 / √7) * (√7 / √7)
This gives us:
tan(A) = -3√7 / 7
This is the exact value of tan(A). Rationalizing the denominator is a common practice in mathematics to simplify expressions and make them easier to work with. It involves removing any square roots from the denominator of a fraction, which helps in avoiding irrational numbers in the denominator.
Rounding to Ten-Thousandths
Okay, we have the exact value: tan(A) = -3√7 / 7. But the question asks us to round to ten-thousandths. So, we need to use a calculator to get the decimal approximation.
Using a calculator, we find:
tan(A) ≈ -1.133893419
Rounding this to ten-thousandths (four decimal places), we get:
tan(A) ≈ -1.1339
So, there you have it! The value of tan(A) in Quadrant II, rounded to ten-thousandths, is approximately -1.1339. Rounding is a crucial skill in many areas of mathematics and science, especially when dealing with approximations or measurements. It ensures that the answer is presented with an appropriate level of precision, avoiding unnecessary complexity while maintaining accuracy.
Conclusion: Mastering Trigonometric Identities
Woohoo! We successfully found tan(A) using the given information and trigonometric identities. This problem perfectly illustrates how important it is to understand and apply these identities, especially the Pythagorean identity and the definition of tangent. Remember, also, that the quadrant plays a vital role in determining the signs of the trigonometric functions.
Trigonometric identities are the backbone of trigonometry, allowing us to simplify expressions, solve equations, and understand the relationships between different trigonometric functions. Mastering these identities is essential for anyone studying mathematics, physics, or engineering. By understanding these concepts thoroughly, you'll be able to tackle a wide range of trigonometric problems with confidence. Keep practicing, and you'll become a trig whiz in no time!
