Exact Trigonometric Function Values Given Cosine

Given the cosine of an angle $\theta$, $\cos \theta = -\frac{4\sqrt{3}}{7}$, this article will walk through the process of finding the exact values of the other trigonometric functions: $\sin \theta$, $\tan \theta$, $\csc \theta$, $\sec \theta$, and $\cot \theta$. This involves using trigonometric identities, understanding the signs of trigonometric functions in different quadrants, and simplifying radical expressions. Let's explore step by step how to derive these values.

Understanding the Given Information

First, let's emphasize the information given: $\cos \theta = -\frac{4\sqrt{3}}{7}$. This tells us that the cosine of the angle $\theta$ is negative. Understanding the sign of trigonometric functions is crucial because it helps determine the quadrant in which the angle $\theta$ lies. Cosine is negative in the second and third quadrants. To proceed further, we need to find the sine of $\theta$, as it is essential for deriving other trigonometric functions. We will employ the Pythagorean identity, a fundamental concept in trigonometry, to calculate $\sin \theta$. The Pythagorean identity is expressed as $\sin^2 \theta + \cos^2 \theta = 1$. By substituting the given value of $\cos \theta$ into this identity, we can solve for $\sin \theta$. This involves algebraic manipulation and simplification, ensuring we account for both positive and negative square roots, as the sine function can be positive or negative depending on the quadrant. The quadrant determination is critical in selecting the correct sign for $\sin \theta$, which directly impacts the values of the other trigonometric functions. This initial step of understanding the given information and setting the stage for further calculations is vital for accurately determining the other function values.

Calculating Sine Using the Pythagorean Identity

To find $\sin \theta$, we utilize the Pythagorean identity: $\sin^2 \theta + \cos^2 \theta = 1$. Given that $\cos \theta = -\frac{4\sqrt{3}}{7}$, we substitute this value into the identity: $\sin^2 \theta + \left(-\frac{4\sqrt{3}}{7}\right)^2 = 1$. Squaring the cosine value, we get $\left(-\frac{4\sqrt{3}}{7}\right)^2 = \frac{16 \cdot 3}{49} = \frac{48}{49}$. Now the equation becomes $\sin^2 \theta + \frac{48}{49} = 1$. To isolate $\sin^2 \theta$, we subtract $\frac{48}{49}$ from both sides: $\sin^2 \theta = 1 - \frac{48}{49} = \frac{49}{49} - \frac{48}{49} = \frac{1}{49}$. Taking the square root of both sides gives us $\sin \theta = \pm \sqrt{\frac{1}{49}} = \pm \frac{1}{7}$. This result indicates that $\sin \theta$ can be either positive or negative, depending on the quadrant in which $\theta$ lies. Since $\cos \theta$ is negative, $\theta$ is in either the second or third quadrant. In the second quadrant, sine is positive, while in the third quadrant, sine is negative. Without additional information to specify the quadrant, we consider both possibilities. However, for the sake of providing a complete solution, we'll proceed assuming $\theta$ is in the second quadrant, where $\sin \theta$ is positive. Thus, we take $\sin \theta = \frac{1}{7}$.

Determining Tangent and Cotangent

With both $\sin \theta$ and $\cos \theta$ known, we can determine $\tan \theta$ and $\cot \theta$. Tangent, defined as $\tan \theta = \frac{\sin \theta}{\cos \theta}$, can be calculated by substituting the values we found: $\tan \theta = \frac{\frac{1}{7}}{-\frac{4\sqrt{3}}{7}}$. To simplify this complex fraction, we multiply the numerator and denominator by 7, resulting in $\tan \theta = \frac{1}{-4\sqrt{3}}$. To rationalize the denominator, we multiply both the numerator and denominator by $\sqrt{3}$: $\tan \theta = \frac{1}{-4\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{-4 \cdot 3} = -\frac{\sqrt{3}}{12}$. Thus, the tangent of $\theta$ is $-\frac{\sqrt{3}}{12}$.

Cotangent, being the reciprocal of tangent, is defined as $\cot \theta = \frac{1}{\tan \theta}$ or equivalently $\cot \theta = \frac{\cos \theta}{\sin \theta}$. Using the former definition and the calculated value of $\tan \theta$, we have $\cot \theta = \frac{1}{-\frac{\sqrt{3}}{12}}$. Simplifying this, we get $\cot \theta = -\frac{12}{\sqrt{3}}$. Rationalizing the denominator by multiplying both the numerator and denominator by $\sqrt{3}$, we obtain $\cot \theta = -\frac{12\sqrt{3}}{3} = -4\sqrt{3}$. Therefore, the cotangent of $\theta$ is $-4\sqrt{3}$. These calculations illustrate how fundamental trigonometric identities and algebraic manipulations are used to find the values of tangent and cotangent once sine and cosine are known.

Finding Secant and Cosecant

To complete the set of trigonometric function values, we need to find $\sec \theta$ and $\csc \theta$. Secant is the reciprocal of cosine, so $\sec \theta = \frac{1}{\cos \theta}$. Given $\cos \theta = -\frac{4\sqrt{3}}{7}$, we have $\sec \theta = \frac{1}{-\frac{4\sqrt{3}}{7}}$. Simplifying this, we get $\sec \theta = -\frac{7}{4\sqrt{3}}$. To rationalize the denominator, we multiply the numerator and denominator by $\sqrt{3}$: $\sec \theta = -\frac{7\sqrt{3}}{4 \cdot 3} = -\frac{7\sqrt{3}}{12}$. Thus, the secant of $\theta$ is $-\frac{7\sqrt{3}}{12}$.

Cosecant is the reciprocal of sine, so $\csc \theta = \frac{1}{\sin \theta}$. We found $\sin \theta = \frac{1}{7}$, so $\csc \theta = \frac{1}{\frac{1}{7}} = 7$. Therefore, the cosecant of $\theta$ is 7. By understanding the reciprocal relationships between trigonometric functions and applying algebraic simplification techniques, we have successfully determined the values of secant and cosecant. These final calculations provide a complete picture of all six trigonometric function values for the given angle $\theta$.

Summary of Trigonometric Function Values

In summary, given $\cos \theta = -\frac{4\sqrt{3}}{7}$, and assuming $\theta$ is in the second quadrant, we have calculated the other trigonometric function values as follows:

• $\sin \theta = \frac{1}{7}$
• $\tan \theta = -\frac{\sqrt{3}}{12}$
• $\cot \theta = -4\sqrt{3}$
• $\sec \theta = -\frac{7\sqrt{3}}{12}$
• $\csc \theta = 7$

These values represent a complete set of trigonometric function values for the angle $\theta$. Understanding and applying trigonometric identities, along with careful algebraic manipulation, is crucial for solving such problems. This step-by-step approach ensures accuracy and clarity in determining all trigonometric function values from a given cosine value. It is also important to remember that the quadrant in which the angle lies significantly impacts the signs of the trigonometric functions, and this must be considered for a complete and correct solution.