Finding Cosecant And Tangent Given Cosine In Quadrant II

In trigonometry, determining the exact values of trigonometric functions for a given angle is a fundamental skill. This article delves into the process of finding the exact values of $\csc \theta$ and $\tan \theta$ when given the cosine of an angle $\theta$ in the second quadrant. Understanding the relationships between trigonometric functions and their signs in different quadrants is crucial for solving such problems. We will explore the use of the Pythagorean identity and the definitions of trigonometric functions to arrive at the solutions.

Understanding the Problem

The problem states that we have an angle $\theta$ located in quadrant II, and we know that $\cos \theta = -\frac{7}{8}$. Our goal is to find the exact values of $\csc \theta$ and $\tan \theta$. To achieve this, we need to utilize trigonometric identities and understand the properties of trigonometric functions in different quadrants. The quadrant in which the angle lies is crucial because it determines the signs of the trigonometric functions. In quadrant II, sine ($\sin$) and its reciprocal, cosecant ($\csc$), are positive, while cosine ($\cos$), tangent ($\tan$), and their reciprocals are negative. This knowledge will help us determine the correct signs for our final answers. We will start by using the Pythagorean identity to find the value of $\sin \theta$, and then we can use the definitions of $\csc \theta$ and $\tan \theta$ in terms of $\sin \theta$ and $\cos \theta$ to find their exact values. This step-by-step approach will ensure accuracy and clarity in our solution. Remember, understanding the underlying principles and relationships is key to mastering trigonometry.

Utilizing the Pythagorean Identity

The Pythagorean identity, $\sin^2 \theta + \cos^2 \theta = 1$, is a cornerstone of trigonometry and allows us to relate sine and cosine. Since we know $\cos \theta = -\frac{7}{8}$, we can substitute this value into the identity to solve for $\sin \theta$. This gives us: $\sin^2 \theta + \left(-\frac{7}{8}\right)^2 = 1$. Simplifying, we get $\sin^2 \theta + \frac{49}{64} = 1$. To isolate $\sin^2 \theta$, we subtract $\frac{49}{64}$ from both sides: $\sin^2 \theta = 1 - \frac{49}{64}$. This simplifies to $\sin^2 \theta = \frac{64}{64} - \frac{49}{64} = \frac{15}{64}$. Now, we take the square root of both sides to find $\sin \theta$: $\sin \theta = \pm\sqrt{\frac{15}{64}} = \pm\frac{\sqrt{15}}{8}$. Since $\theta$ is in quadrant II, where sine is positive, we choose the positive root: $\sin \theta = \frac{\sqrt{15}}{8}$. This step is crucial as it provides us with the value of sine, which we will use to find the other trigonometric functions. The correct application of the Pythagorean identity and the consideration of the quadrant's sign are essential for obtaining the correct value of $\sin \theta$. With $\sin \theta$ determined, we can now move on to finding the values of $\csc \theta$ and $\tan \theta$.

Finding the Exact Value of $\csc \theta$

The cosecant function ($\csc \theta$) is defined as the reciprocal of the sine function, i.e., $\csc \theta = \frac{1}{\sin \theta}$. We have already found that $\sin \theta = \frac{\sqrt{15}}{8}$ for the given angle $\theta$ in quadrant II. Therefore, to find $\csc \theta$, we simply take the reciprocal of $\sin \theta$: $\csc \theta = \frac{1}{\frac{\sqrt{15}}{8}}$. This simplifies to $\csc \theta = \frac{8}{\sqrt{15}}$. To rationalize the denominator, we multiply both the numerator and the denominator by $\sqrt{15}$: $\csc \theta = \frac{8\sqrt{15}}{15}$. This is the exact value of $\csc \theta$. The process of finding the cosecant involves a straightforward application of the reciprocal relationship with sine. Ensuring the denominator is rationalized is a standard practice in expressing trigonometric values. The positive value of $\csc \theta$ aligns with the fact that cosecant is positive in quadrant II, further validating our result. Now that we have found the exact value of $\csc \theta$, we can proceed to determine the exact value of $\tan \theta$ using the values of $\sin \theta$ and $\cos \theta$.

Determining the Exact Value of $\tan \theta$

The tangent function ($\tan \theta$) is defined as the ratio of the sine function to the cosine function, i.e., $\tan \theta = \frac{\sin \theta}{\cos \theta}$. We have already determined that $\sin \theta = \frac{\sqrt{15}}{8}$ and we are given that $\cos \theta = -\frac{7}{8}$. Substituting these values into the definition of tangent, we get: $\tan \theta = \frac{\frac{\sqrt{15}}{8}}{-\frac{7}{8}}$. To simplify this complex fraction, we multiply the numerator by the reciprocal of the denominator: $\tan \theta = \frac{\sqrt{15}}{8} \times \left(-\frac{8}{7}\right)$. This simplifies to $\tan \theta = -\frac{\sqrt{15}}{7}$. This is the exact value of $\tan \theta$. The negative sign is consistent with the fact that tangent is negative in quadrant II. The calculation involves a straightforward application of the definition of tangent and simplification of the resulting fraction. The correct determination of the sign is crucial, and it aligns with the quadrant in which the angle lies. With the exact values of both $\csc \theta$ and $\tan \theta$ found, we have successfully addressed the problem. This process demonstrates the importance of understanding trigonometric identities and the properties of trigonometric functions in different quadrants.

Conclusion

In conclusion, we have successfully found the exact values of $\csc \theta$ and $\tan \theta$ given that $\cos \theta = -\frac{7}{8}$ and $\theta$ is in quadrant II. By utilizing the Pythagorean identity and the definitions of cosecant and tangent, we determined that $\csc \theta = \frac{8\sqrt{15}}{15}$ and $\tan \theta = -\frac{\sqrt{15}}{7}$. This exercise highlights the importance of understanding the relationships between trigonometric functions and their signs in different quadrants. The step-by-step approach, involving the use of the Pythagorean identity to find sine, the reciprocal relationship to find cosecant, and the ratio of sine to cosine to find tangent, is crucial for solving such problems accurately. Furthermore, the consideration of the quadrant in which the angle lies is essential for determining the correct signs of the trigonometric functions. This problem serves as a valuable illustration of how trigonometric identities and principles can be applied to find the exact values of trigonometric functions, a fundamental skill in trigonometry and calculus.