Finding Cos(A) In Quadrant II Given Sin(A) = 1/4

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In trigonometry, we often encounter scenarios where we need to find the values of other trigonometric functions given the value of one function. A fundamental identity that helps us in this pursuit is the Pythagorean identity: $\sin^2(A) + \cos^2(A) = 1$. This identity is derived from the Pythagorean theorem and holds true for all angles A. In this article, we will explore how to use this identity to find $\cos(A)$ when given $\sin(A) = \frac{1}{4}$ and the angle A lies in the second quadrant. Understanding the implications of the quadrant in which the angle lies is crucial because it determines the signs of the trigonometric functions.

Understanding the Problem

We are given that $\sin(A) = \frac{1}{4}$. The sine function represents the ratio of the opposite side to the hypotenuse in a right-angled triangle. In the unit circle, it corresponds to the y-coordinate of a point on the circle. Since $\\sin(A)$ is positive, the angle A must lie in either the first or second quadrant because sine is positive in these quadrants. However, we are specifically told that A is in the second quadrant. This piece of information is vital because it tells us that the x-coordinate, and hence the cosine of the angle, will be negative. The cosine function represents the ratio of the adjacent side to the hypotenuse, and in the unit circle, it corresponds to the x-coordinate of a point on the circle. In the second quadrant, x-coordinates are negative.

To find $\cos(A)$, we will use the Pythagorean identity: $\sin^2(A) + \cos^2(A) = 1$. This identity is a cornerstone of trigonometry and is derived directly from the Pythagorean theorem applied to the unit circle. By substituting the given value of $\\sin(A)$ into this identity, we can solve for $\cos(A)$. It is essential to remember that the value of $\cos(A)$ will be negative in the second quadrant. This is because in the second quadrant, the x-coordinate of any point on the unit circle is negative, and cosine corresponds to the x-coordinate.

Applying the Pythagorean Identity

To find $\cos(A)$, we start with the Pythagorean identity:

sin⁡2(A)+cos⁡2(A)=1\sin^2(A) + \cos^2(A) = 1

We are given that $\sin(A) = \frac{1}{4}$. Substituting this value into the identity, we get:

(14)2+cos⁡2(A)=1\left(\frac{1}{4}\right)^2 + \cos^2(A) = 1

116+cos⁡2(A)=1\frac{1}{16} + \cos^2(A) = 1

Now, we need to isolate $\cos^2(A)$. To do this, we subtract $\frac{1}{16}$ from both sides of the equation:

cos⁡2(A)=1−116\cos^2(A) = 1 - \frac{1}{16}

cos⁡2(A)=1616−116\cos^2(A) = \frac{16}{16} - \frac{1}{16}

cos⁡2(A)=1516\cos^2(A) = \frac{15}{16}

Next, we take the square root of both sides to solve for $\cos(A)$:

cos⁡(A)=±1516\cos(A) = \pm\sqrt{\frac{15}{16}}

cos⁡(A)=±154\cos(A) = \pm\frac{\sqrt{15}}{4}

Since angle A is in the second quadrant, where cosine is negative, we choose the negative root:

cos⁡(A)=−154\cos(A) = -\frac{\sqrt{15}}{4}

Calculating the Numerical Value

Now that we have found the exact value of $\cos(A)$, which is $- \frac{\sqrt{15}}{4}$, we need to calculate its numerical value rounded to ten-thousandths. We know that $\sqrt{15}$ is approximately 3.87298. Therefore, we can substitute this value into the expression for $\cos(A)$:

cos⁡(A)=−3.872984\cos(A) = -\frac{3.87298}{4}

Now, we divide 3.87298 by 4:

cos⁡(A)≈−0.968245\cos(A) \approx -0.968245

We are asked to round this value to ten-thousandths, which means we need to round to four decimal places. Looking at the fifth decimal place (5), we see that it is 5 or greater, so we round up the fourth decimal place:

cos⁡(A)≈−0.9682\cos(A) \approx -0.9682

Thus, the value of $\cos(A)$ rounded to ten-thousandths is -0.9682.

Significance of the Quadrant

The quadrant in which the angle A lies plays a crucial role in determining the sign of the trigonometric functions. In the unit circle, the quadrants are defined as follows:

  • Quadrant I: 0° to 90° (0 to $\frac{\pi}{2}$ radians) – All trigonometric functions are positive.
  • Quadrant II: 90° to 180° ($\frac{\pi}{2}$ to $\pi$ radians) – Sine (sin⁡\sin) and cosecant (csc⁡\csc) are positive; cosine (cos⁡\cos), tangent (tan⁡\tan), secant (sec⁡\sec), and cotangent (cot⁡\cot) are negative.
  • Quadrant III: 180° to 270° ($\pi$ to $\frac{3\pi}{2}$ radians) – Tangent (tan⁡\tan) and cotangent (cot⁡\cot) are positive; sine (sin⁡\sin), cosine (cos⁡\cos), cosecant (csc⁡\csc), and secant (sec⁡\sec) are negative.
  • Quadrant IV: 270° to 360° ($\frac{3\pi}{2}$ to $2\pi$ radians) – Cosine (cos⁡\cos) and secant (sec⁡\sec) are positive; sine (sin⁡\sin), tangent (tan⁡\tan), cosecant (csc⁡\csc), and cotangent (cot⁡\cot) are negative.

In our case, since angle A is in the second quadrant, we know that sine is positive and cosine is negative. This is why we chose the negative square root when solving for $\cos(A)$. If we had not considered the quadrant, we would have obtained two possible values for $\cos(A)$, and we would not have been able to determine the correct one.

Alternative Methods and Insights

While the Pythagorean identity is the most straightforward method to solve this problem, there are other ways to approach it. One could use the unit circle definition of trigonometric functions. Since we know that $\sin(A) = \frac{1}{4}$, we can visualize a point on the unit circle with a y-coordinate of $\\frac{1}{4}$. Because A is in the second quadrant, this point will have a negative x-coordinate. We can then use the equation of the unit circle, $x^2 + y^2 = 1$, to solve for the x-coordinate, which corresponds to $\cos(A)$.

Another insightful approach is to consider the right-angled triangle formed by the point on the unit circle, the origin, and the x-axis. The sine of the angle is the ratio of the opposite side to the hypotenuse, and the cosine is the ratio of the adjacent side to the hypotenuse. By applying the Pythagorean theorem to this triangle, we can find the length of the adjacent side and then determine the cosine of the angle. Remember to consider the sign of the cosine based on the quadrant.

Common Mistakes to Avoid

When solving trigonometric problems, several common mistakes can lead to incorrect answers. One frequent mistake is forgetting to consider the quadrant of the angle. As we have seen, the quadrant determines the signs of the trigonometric functions, and neglecting this can result in choosing the wrong sign for the answer. Another common mistake is making errors in algebraic manipulations, such as incorrectly isolating variables or taking square roots. It is crucial to double-check each step to ensure accuracy.

Additionally, students sometimes confuse the trigonometric identities or apply them incorrectly. It is essential to have a solid understanding of the fundamental identities and their derivations. Regular practice and problem-solving can help reinforce these concepts and prevent mistakes. For example, when using the Pythagorean identity, ensure that you are substituting the values correctly and that you are solving for the correct trigonometric function.

Conclusion

In summary, we have successfully found $\cos(A)$ given that $\sin(A) = \frac{1}{4}$ and A is in the second quadrant. We used the Pythagorean identity, $\sin^2(A) + \cos^2(A) = 1$, to solve for $\cos(A)$. By substituting the given value of $\\sin(A)$ and considering that cosine is negative in the second quadrant, we found that $\cos(A) = -\frac{\sqrt{15}}{4}$. We then calculated the numerical value of $\cos(A)$ to be approximately -0.9682 when rounded to ten-thousandths.

This problem illustrates the importance of understanding trigonometric identities and the significance of the quadrant in determining the signs of trigonometric functions. By mastering these concepts, you can confidently solve a wide range of trigonometric problems. Remember to always consider the given information carefully and choose the appropriate methods and identities to arrive at the correct solution. Consistent practice and a thorough understanding of the fundamentals are key to success in trigonometry.