Question

One function gives all six Given the following information about one trigonometric function, evaluate the other five functions. $$\csc \theta=\frac{13}{12} \text { and } 0<\theta<\pi / 2$$

Short answer

Question: Given that \(\csc \theta = \frac{13}{12}\) and \(0 < \theta < \frac{\pi}{2}\), find all six trigonometric functions for angle θ. Answer: 1. \(\sin \theta = \frac{12}{13}\) 2. \(\cos \theta = \frac{5}{13}\) 3. \(\tan \theta = \frac{12}{5}\) 4. \(\csc \theta = \frac{13}{12}\) 5. \(\sec \theta = \frac{13}{5}\) 6. \(\cot \theta = \frac{5}{12}\)

Step-by-step solution

Find the value of the sine function

Since we are given that \(\csc \theta = \frac{13}{12}\), we can find the value of the sine function by taking the reciprocal of the cosecant function: $$\sin \theta = \frac{1}{\csc \theta} = \frac{1}{\frac{13}{12}} = \frac{12}{13}$$

Find the value of the cosine function

To find the cosine function's value, we can use the Pythagorean identity: \(\sin^2\theta + \cos^2\theta = 1\). Then, substituting \(\sin \theta = \frac{12}{13}\): $$\left(\frac{12}{13}\right)^2 + \cos^2\theta = 1$$ Solve for \(\cos \theta\): $$\cos^2\theta = 1 - \left(\frac{12}{13}\right)^2 = \frac{25}{169}$$ Take the square root of both sides given that 0 < θ < π / 2, thus cosine is positive: $$\cos \theta = \sqrt{\frac{25}{169}} = \frac{5}{13}$$

Find the secant, cosecant, and cotangent

Now that we have sine and cosine values, we can calculate the other three functions. Secant (reciprocal of cosine): $$\sec \theta = \frac{1}{\cos \theta} = \frac{1}{\frac{5}{13}}= \frac{13}{5}$$ Cotangent (reciprocal of tangent): $$\cot \theta = \frac{1}{\tan \theta} = \frac{1}{\frac{\sin \theta}{\cos \theta}} = \frac{\cos \theta}{\sin \theta} = \frac{\frac{5}{13}}{\frac{12}{13}}= \frac{5}{12}$$ Now we have found all six trigonometric functions for the given angle θ: 1. \(\sin \theta = \frac{12}{13}\) 2. \(\cos \theta = \frac{5}{13}\) 3. \(\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{12}{5}\) 4. \(\csc \theta = \frac{13}{12}\) (given) 5. \(\sec \theta = \frac{13}{5}\) 6. \(\cot \theta = \frac{5}{12}\)

Key concepts

Cosecant Function

The cosecant function, denoted as \( \csc \theta \), is one of the six fundamental trigonometric functions. It is specifically the reciprocal of the sine function. In simpler terms, if you know the value of \( \sin \theta \), you can figure out \( \csc \theta \) by just taking its reciprocal. Mathematically, this is represented as \( \csc \theta = \frac{1}{\sin \theta} \).
Understanding this relationship is crucial, especially when you're given \( \csc \theta \) and need to find \( \sin \theta \). In this exercise, knowing \( \csc \theta = \frac{13}{12} \) helped us to easily derive \( \sin \theta = \frac{12}{13} \) by flipping the fraction. Similarly, when you need to solve problems involving cosecant, always remember the connection to the sine function this reciprocal nature reveals.
This reciprocal identification is a part of a broader group known as the Reciprocal Identities, which we'll explore later.

Pythagorean Identity

The Pythagorean identity is one of the cornerstones of trigonometry. It defines a relation among the sine, cosine, and tangent functions, providing a powerful tool in deriving unknowns based on known trigonometric values. For sine and cosine, this identity is expressed as \( \sin^2 \theta + \cos^2 \theta = 1 \).
In our context, we knew \( \sin \theta = \frac{12}{13} \) thanks to the cosecant reciprocal property. Substituting this into the Pythagorean identity, \( (\frac{12}{13})^2 + \cos^2 \theta = 1 \), allows us to solve for \( \cos \theta \). This crucial step was what helped us arrive at \( \cos \theta = \frac{5}{13} \).
The beauty of the Pythagorean identity is its ability to help us find missing trigonometric values by algebraic means. Anytime you know one function, say sine or cosine, this identity is your express ticket to finding its counterpart.

Reciprocal Identities

Reciprocal identities are fundamental in trigonometry, granting the means to easily switch between certain trigonometric functions. We already explored the cosecant's relationship to sine with \( \csc \theta = \frac{1}{\sin \theta} \). Similarly, secant and cosine share this reciprocal connection, expressed as \( \sec \theta = \frac{1}{\cos \theta} \), and \( \cos \theta = \frac{1}{\sec \theta} \).
The last member of this reciprocal triad combines tangent and cotangent. You have \( \tan \theta = \frac{1}{\cot \theta} \) and \( \cot \theta = \frac{1}{\tan \theta} \), but also \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), tying them back to sine and cosine.
In the exercise, knowing \( \cos \theta \) allowed us to determine \( \sec \theta = \frac{13}{5} \). Similarly, by finding \( \tan \theta = \frac{12}{5} \), we concluded \( \cot \theta = \frac{5}{12} \). These identities simplify the process of finding all six trigonometric functions when one is known.