Finding Cos(x) Tan(x) And Cos(x - 3π/2) Given Sin(x) = 7/25 And Π/2 < X < Π

In this article, we'll explore how to find the values of other trigonometric functions given the sine of an angle and the quadrant in which the angle lies. Specifically, we are given that $\sin(x) = \frac{7}{25}$ and $\frac{\pi}{2} < x < \pi$. Our goal is to find $\cos(x)$, $\tan(x)$, and $\cos(x - \frac{3\pi}{2})$. This problem combines trigonometric identities and an understanding of the unit circle, offering a comprehensive approach to solving trigonometric problems.

a. Finding cos(x)

To determine the value of cos(x), we can use the fundamental Pythagorean trigonometric identity, which states that $\sin^2(x) + \cos^2(x) = 1$. This identity is a cornerstone of trigonometry, derived directly from the Pythagorean theorem applied to the unit circle. By substituting the given value of $\sin(x)$ into this identity, we can solve for $\cos(x)$.

We are given that $\sin(x) = \frac{7}{25}$. Substituting this into the Pythagorean identity, we get:

$$\left(\frac{7}{25}\right)^2 + \cos^2(x) = 1$$

$$\frac{49}{625} + \cos^2(x) = 1$$

Now, we isolate $\cos^2(x)$:

$$\cos^2(x) = 1 - \frac{49}{625}$$

$$\cos^2(x) = \frac{625 - 49}{625}$$

$$\cos^2(x) = \frac{576}{625}$$

Taking the square root of both sides, we get:

$$\cos(x) = \pm\sqrt{\frac{576}{625}}$$

$$\cos(x) = \pm\frac{24}{25}$$

At this stage, we have two possible values for $\cos(x)$: $\frac{24}{25}$ and $-\frac{24}{25}$. To determine the correct sign, we must consider the quadrant in which the angle x lies. We are given that $\frac{\pi}{2} < x < \pi$. This inequality tells us that x is in the second quadrant. In the second quadrant, the cosine function is negative. This is because, on the unit circle, the x-coordinate (which corresponds to the cosine value) is negative in the second quadrant.

Therefore, we choose the negative value for $\cos(x)$:

$$\cos(x) = -\frac{24}{25}$$

In summary, to find $\cos(x)$, we utilized the Pythagorean identity and the information about the quadrant in which x lies. The identity allowed us to find the possible values, and the quadrant information helped us select the correct sign. Understanding these principles is crucial for solving various trigonometric problems.

b. Finding tan(x)

Now, let's determine the value of tan(x). The tangent function is defined as the ratio of the sine function to the cosine function: $\tan(x) = \frac{\sin(x)}{\cos(x)}$. This relationship is fundamental in trigonometry and allows us to find $\tan(x)$ if we know $\sin(x)$ and $\cos(x)$. We have already been given the value of $\sin(x)$ and have calculated the value of $\cos(x)$ in the previous section. So, we can directly use these values to find $\tan(x)$.

We know that $\sin(x) = \frac{7}{25}$ and $\cos(x) = -\frac{24}{25}$. Substituting these values into the definition of the tangent function, we get:

$$\tan(x) = \frac{\frac{7}{25}}{-\frac{24}{25}}$$

To simplify this expression, we can multiply both the numerator and the denominator by 25. This effectively cancels out the denominators in the fractions:

$$\tan(x) = \frac{7}{-24}$$

$$\tan(x) = -\frac{7}{24}$$

Therefore, the value of $\tan(x)$ is $-\frac{7}{24}$. It's important to note the negative sign, which arises from the fact that $\tan(x)$ is negative in the second quadrant, where our angle x lies. The tangent function is positive in the first and third quadrants and negative in the second and fourth quadrants. Understanding the signs of trigonometric functions in different quadrants is crucial for accurately solving trigonometric problems.

In this part, we calculated $\tan(x)$ using the definition $\tan(x) = \frac{\sin(x)}{\cos(x)}$ and the values we already had for $\sin(x)$ and $\cos(x)$. This straightforward calculation highlights the importance of knowing the fundamental trigonometric identities and the signs of the trigonometric functions in different quadrants.

c. Finding cos(x - 3π/2)

Finally, let's find the value of cos(x - \frac{3\pi}{2}). To do this, we will use the cosine subtraction formula, which is a crucial trigonometric identity. The formula states that:

$$\cos(A - B) = \cos(A)\cos(B) + \sin(A)\sin(B)$$

In our case, A = x and B = $\frac{3\pi}{2}$. So, we need to find $\cos(\frac{3\pi}{2})$ and $\sin(\frac{3\pi}{2})$.

First, let's consider the angle $\frac{3\pi}{2}$. On the unit circle, this angle corresponds to the point (0, -1). The x-coordinate of this point gives us the cosine of the angle, and the y-coordinate gives us the sine of the angle. Therefore:

$$\cos\left(\frac{3\pi}{2}\right) = 0$$

$$\sin\left(\frac{3\pi}{2}\right) = -1$$

Now, we can substitute these values, along with the given $\sin(x) = \frac{7}{25}$ and the calculated $\cos(x) = -\frac{24}{25}$, into the cosine subtraction formula:

$$\cos\left(x - \frac{3\pi}{2}\right) = \cos(x)\cos\left(\frac{3\pi}{2}\right) + \sin(x)\sin\left(\frac{3\pi}{2}\right)$$

$$\cos\left(x - \frac{3\pi}{2}\right) = \left(-\frac{24}{25}\right)(0) + \left(\frac{7}{25}\right)(-1)$$

$$\cos\left(x - \frac{3\pi}{2}\right) = 0 - \frac{7}{25}$$

$$\cos\left(x - \frac{3\pi}{2}\right) = -\frac{7}{25}$$

Thus, the value of $\cos(x - \frac{3\pi}{2})$ is $-\frac{7}{25}$. This calculation demonstrates the power of trigonometric identities in simplifying complex expressions. The cosine subtraction formula allowed us to break down the problem into manageable parts and use the values we already knew to find the final result. In addition to the cosine subtraction formula, it's crucial to understand the unit circle and the values of trigonometric functions at key angles, such as $\frac{3\pi}{2}$. This knowledge, combined with the appropriate identities, enables us to solve a wide range of trigonometric problems efficiently.

Conclusion

In this article, we successfully found the values of $\cos(x)$, $\tan(x)$, and $\cos(x - \frac{3\pi}{2})$ given that $\sin(x) = \frac{7}{25}$ and $\frac{\pi}{2} < x < \pi$. We used the Pythagorean identity, the definition of the tangent function, the cosine subtraction formula, and the unit circle to arrive at our solutions. This problem serves as a great example of how different trigonometric concepts and identities can be combined to solve a multi-part problem. Understanding these concepts is essential for anyone studying trigonometry and related fields.