Question

Given the following information about one trigonometric function, evaluate the other five functions. $$\sec \theta=\frac{5}{3} \text { and } 3 \pi / 2<\theta<2 \pi$$

Short answer

Given that the secant function is $$\sec \theta = \frac{5}{3}$$ and $$3 \pi / 2 < \theta < 2 \pi$$, find the remaining trigonometric functions. Solution: 1. Cosine function: $$\cos \theta = \frac{3}{5}$$ 2. Sine function: $$\sin \theta = -\frac{4}{5}$$ 3. Tangent function: $$\tan \theta = -\frac{4}{3}$$ 4. Cosecant function: $$\csc \theta = -\frac{5}{4}$$ 5. Cotangent function: $$\cot \theta = -\frac{3}{4}$$

Step-by-step solution

Find the Cosine Function

Recall the relationship between the secant and cosine functions: $$\sec \theta = \frac{1}{\cos \theta}$$. Since we are given that $$\sec \theta = \frac{5}{3}$$, we can use this relationship to find the value of $$\cos \theta$$: $$\cos \theta = \frac{1}{\sec \theta} = \frac{1}{\frac{5}{3}} = \frac{3}{5}$$.

Find the Sine Function

Since we know the cosine function, we can use the Pythagorean theorem for trigonometric functions, which states that: $$\sin^2 \theta + \cos^2 \theta = 1$$ Substitute the value of $$\cos \theta$$ we found earlier: $$\sin^2 \theta + \left(\frac{3}{5}\right)^2 = 1$$ Now, solve for $$\sin \theta$$: $$\sin^2 \theta = 1 - \left(\frac{3}{5}\right)^2 = 1 - \frac{9}{25} = \frac{16}{25}$$ Take the square root to find the sine function. Since the angle $$\theta$$ is in the range $$3 \pi / 2 < \theta < 2 \pi$$, the sine function will be negative: $$\sin \theta = -\sqrt{\frac{16}{25}} = -\frac{4}{5}$$.

Find the Tangent Function

We can calculate the tangent function using the definitions of sine and cosine: $$\tan \theta = \frac{\sin \theta}{\cos \theta}$$ Substitute the values we found for $$\sin \theta$$ and $$\cos \theta$$: $$\tan \theta = \frac{-\frac{4}{5}}{\frac{3}{5}} = -\frac{4}{3}$$

Find the Cosecant and Cotangent Functions

Since we have already calculated the sine, cosine, and tangent functions, we can easily find the remaining functions: 1. Cosecant ($$\csc \theta$$): Inverse of sine function: $$\csc \theta = \frac{1}{\sin \theta} = \frac{1}{-\frac{4}{5}} = -\frac{5}{4}$$ 2. Cotangent ($$\cot \theta$$): Inverse of tangent function: $$\cot \theta = \frac{1}{\tan \theta} = \frac{1}{-\frac{4}{3}} = -\frac{3}{4}$$ Thus, the remaining trigonometric functions are: $$\sin \theta = -\frac{4}{5}, \cos \theta = \frac{3}{5}, \tan \theta = -\frac{4}{3}, \csc \theta = -\frac{5}{4}, \text { and } \cot \theta = -\frac{3}{4}$$

Key concepts

Secant Function

The secant function, denoted as \( \sec \theta \), is one of the six fundamental trigonometric functions. The secant is the reciprocal of the cosine function. This means that it represents the ratio of the hypotenuse to the adjacent side in a right triangle. When you know \( \sec \theta \), you can easily find the cosine using the relationship \( \sec \theta = \frac{1}{\cos \theta} \). In this exercise, with \( \sec \theta = \frac{5}{3} \), we calculated \( \cos \theta \) as \( \frac{3}{5} \).
Understanding the secant function is crucial not only for solving problems involving cosine but also for determining other trigonometric functions through identities and relationships.

Pythagorean Identity

The Pythagorean identity is a fundamental relationship in trigonometry that connects sine and cosine with the number 1. This identity states that \( \sin^2 \theta + \cos^2 \theta = 1 \) for any angle \( \theta \). It is incredibly useful for finding one function when you have the other. In our scenario, we determined \( \sin \theta \) using this identity:

• We substituted the known cosine \( \left(\cos \theta = \frac{3}{5}\right) \) into the identity.
• Solved for \( \sin^2 \theta = \frac{16}{25} \).
• Then took the square root, keeping in mind the quadrant sign, resulting in \( \sin \theta = -\frac{4}{5} \).

This identity not only provides important connections among trigonometric functions but also aids in solving more complex trigonometric equations.

Trigonometric Identities

Trigonometric identities are equations that hold true for all angle measures. They provide essential relationships between the six trigonometric functions: sine, cosine, tangent, cosecant, secant, and cotangent. These identities allow simplifies of expressions and solving of equations in trigonometry.
Some of the vital identities include:

• The reciprocal identities, like \( \sec \theta = \frac{1}{\cos \theta} \) and \( \csc \theta = \frac{1}{\sin \theta} \).
• The quotient identities, such as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) and \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).
• Pythagorean identities, including variations like \( 1 + \tan^2 \theta = \sec^2 \theta \).

These identities are essential for understanding the comprehensive picture of trigonometry and solving trigonometric problems efficiently.

Angle Measures

The measure of angles in trigonometry is often given in radians or degrees, with radians being the standard unit in mathematical contexts. Understanding where an angle lies within these units—particularly in which quadrant—helps in determining the sign (positive or negative) of trigonometric functions.
For instance, in this problem, \( \theta \) lies between \( \frac{3\pi}{2} \) and \( 2\pi \) radians, which is in the fourth quadrant. Knowing this is critical because:

• Sine and tangent functions are negative in the fourth quadrant, guiding us to find \( \sin \theta = -\frac{4}{5} \) and \( \tan \theta = -\frac{4}{3} \).
• Cosine, however, is positive, which aligns with \( \cos \theta = \frac{3}{5} \).

Always consider the quadrant when calculating trigonometric values to ensure accuracy.