Find Cos(A) Given Sin(A) In Quadrant I

Hey guys! Let's dive into a trigonometric problem where we need to find the value of cos(A) given sin(A) and the quadrant information. This is a classic problem that combines trigonometric identities with understanding the behavior of trigonometric functions in different quadrants. So, let’s break it down step by step.

Problem Statement

We are given that $\sin(A) = \frac{2}{5}$ and we need to find $\cos(A)$ in Quadrant I. We'll be using the Pythagorean trigonometric identity $\sin^2(A) + \cos^2(A) = 1$. This identity is a cornerstone of trigonometry, derived directly from the Pythagorean theorem, and it's super useful for relating sine and cosine.

Step 1: Understanding the Pythagorean Identity

The Pythagorean identity, $\sin^2(A) + \cos^2(A) = 1$, is our starting point. It tells us that for any angle A, the sum of the square of its sine and the square of its cosine is always equal to 1. This identity is incredibly powerful because it allows us to find the cosine of an angle if we know its sine, and vice versa. It's derived from the fundamental relationship in a right-angled triangle, where the square of the hypotenuse is equal to the sum of the squares of the other two sides. Think of sine and cosine as the ratios of these sides, and you'll see how this identity naturally emerges.

In our case, we know $\sin(A)$, so we can plug it into the identity and solve for $\cos(A)$. This is a straightforward algebraic manipulation, but it's crucial to understand why this identity works in the first place. It's not just a formula to memorize; it's a reflection of the underlying geometry of the unit circle and the definitions of sine and cosine.

Step 2: Plugging in the Value of sin(A)

Now, let's substitute the given value of $\sin(A)$ into the identity. We have $\sin(A) = \frac{2}{5}$, so we'll replace $\sin(A)$ in the equation with this value. This gives us:

$\qquad \left(\frac{2}{5}\right)^2 + \cos^2(A) = 1$

This is a simple substitution, but it's a critical step. We're essentially translating the problem from a general trigonometric relationship to a specific equation that we can solve. By squaring the fraction, we're preparing to isolate $\cos^2(A)$ and eventually find $\cos(A)$. It’s like setting up the pieces on a chessboard before making your move. Each step is deliberate and brings us closer to the solution.

Step 3: Simplifying the Equation

Let's simplify the equation further. Squaring $\frac{2}{5}$ gives us $\frac{4}{25}$. So, our equation becomes:

$\qquad \frac{4}{25} + \cos^2(A) = 1$

Now, we need to isolate $\cos^2(A)$. To do this, we'll subtract $\frac{4}{25}$ from both sides of the equation. This is a basic algebraic manipulation, but it's essential to maintain the balance of the equation. What we do on one side, we must do on the other. It's like keeping a scale balanced.

Subtracting $\frac{4}{25}$ from both sides, we get:

$\qquad \cos^2(A) = 1 - \frac{4}{25}$

Step 4: Finding a Common Denominator

To subtract the fractions, we need a common denominator. We can rewrite 1 as $\frac{25}{25}$. This allows us to combine the terms on the right side of the equation:

$\qquad \cos^2(A) = \frac{25}{25} - \frac{4}{25}$

Now we can easily subtract the fractions:

$\qquad \cos^2(A) = \frac{25 - 4}{25} = \frac{21}{25}$

Step 5: Solving for cos(A)

We now have $\cos^2(A) = \frac{21}{25}$. To find $\cos(A)$, we need to take the square root of both sides of the equation:

$\qquad \cos(A) = \pm \sqrt{\frac{21}{25}}$

Remember, when we take the square root, we get both a positive and a negative solution. This is because both the positive and negative square roots, when squared, will give us the same positive value. However, we need to consider the quadrant information to determine the correct sign.

Step 6: Considering the Quadrant

This is where the information about Quadrant I becomes crucial. In Quadrant I, both cosine and sine are positive. This is because in the unit circle, Quadrant I corresponds to angles between 0 and 90 degrees, where both the x-coordinate (cosine) and the y-coordinate (sine) are positive. This is a fundamental concept in trigonometry and understanding it will help you solve many problems.

Since we are in Quadrant I, we know that $\cos(A)$ must be positive. Therefore, we can discard the negative solution and take only the positive square root:

$\qquad \cos(A) = \sqrt{\frac{21}{25}}$

Step 7: Simplifying the Square Root

We can simplify the square root by taking the square root of the numerator and the denominator separately:

$\qquad \cos(A) = \frac{\sqrt{21}}{\sqrt{25}} = \frac{\sqrt{21}}{5}$

This is the exact value of $\cos(A)$. However, the problem asks us to round to the ten-thousandth, so we need to find the decimal approximation.

Step 8: Decimal Approximation

To find the decimal approximation, we can use a calculator to find the square root of 21:

$\qquad \sqrt{21} \approx 4.58257569496$

Now, divide this by 5:

$\qquad \cos(A) = \frac{4.58257569496}{5} \approx 0.91651513899$

Rounding to the ten-thousandth (four decimal places), we get:

$\qquad \cos(A) \approx 0.9165$

Final Answer

So, the value of $\cos(A)$ in Quadrant I, rounded to the ten-thousandth, is approximately 0.9165. We've successfully used the Pythagorean identity and quadrant information to find our answer. Remember, trigonometry is all about relationships between angles and sides, and understanding these relationships is key to solving problems.

Summary of Steps

Let's recap the steps we took to solve this problem:

1. Understand the Pythagorean Identity: $\sin^2(A) + \cos^2(A) = 1$
2. Plug in the Value of sin(A): $\left(\frac{2}{5}\right)^2 + \cos^2(A) = 1$
3. Simplify the Equation: $\frac{4}{25} + \cos^2(A) = 1$ $\rightarrow$ $\cos^2(A) = 1 - \frac{4}{25}$
4. Find a Common Denominator: $\cos^2(A) = \frac{25}{25} - \frac{4}{25} = \frac{21}{25}$
5. Solve for cos(A): $\cos(A) = \pm \sqrt{\frac{21}{25}}$
6. Consider the Quadrant: In Quadrant I, $\cos(A)$ is positive, so $\cos(A) = \sqrt{\frac{21}{25}}$
7. Simplify the Square Root: $\cos(A) = \frac{\sqrt{21}}{5}$
8. Decimal Approximation: $\cos(A) \approx 0.9165$

By following these steps, we were able to find the value of $\cos(A)$ accurately and efficiently. Keep practicing, and you'll become a trig master in no time!

Hope this helps, guys! Let me know if you have any questions.