Question

One function gives all six 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

Question: Given \(\sec \theta = \frac{5}{3}\) and \(3 \pi / 2 < \theta < 2 \pi\), find the values of the other five trigonometric functions. Answer: The other five trigonometric functions are as follows: 1. \(\sin \theta = -\frac{4}{5}\) 2. \(\cos \theta = \frac{3}{5}\) 3. \(\tan \theta = -\frac{4}{3}\) 4. \(\csc \theta = -\frac{5}{4}\) 5. \(\cot \theta = -\frac{3}{4}\)

Step-by-step solution

Determine the sine and cosine functions using the given secant function

Since we know that \(\sec \theta = \frac{5}{3}\), this implies that \(\cos \theta = \frac{1}{\sec \theta} = \frac{1}{5/3} = \frac{3}{5}\). Now, in order to find the sine function, we can use the identity \(\sin^{2} \theta + \cos^{2} \theta = 1\), which means \(\sin^{2} \theta = 1 - \cos^{2} \theta = 1 - (\frac{3}{5})^2 = 1 - \frac{9}{25} = \frac{16}{25}\). Since \(3 \pi / 2 < \theta < 2 \pi\), we are in the fourth quadrant; therefore, sine should be negative, which gives us \(\sin \theta = -\frac{4}{5}.\)

Find the tangent function

To find the tangent function, we can use the relationship \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). Plugging in the values we found in step 1, we get \(\tan \theta = \frac{-\frac{4}{5}}{\frac{3}{5}} = -\frac{4}{3}\).

Find the cosecant function

To find the cosecant function, we can use the relationship \(\csc \theta = \frac{1}{\sin \theta}\). Plugging in the value we found in step 1, we get \(\csc \theta = \frac{1}{-\frac{4}{5}} = -\frac{5}{4}\).

Find the cotangent function

To find the cotangent function, we can use the relationship \(\cot \theta = \frac{1}{\tan \theta}\). Plugging in the value we found in step 2, we get \(\cot \theta = \frac{1}{-\frac{4}{3}} = -\frac{3}{4}\). Now, we have values for all six trigonometric functions: 1. \(\sec \theta = \frac{5}{3}\) 2. \(\sin \theta = -\frac{4}{5}\) 3. \(\cos \theta = \frac{3}{5}\) 4. \(\tan \theta = -\frac{4}{3}\) 5. \(\csc \theta = -\frac{5}{4}\) 6. \(\cot \theta = -\frac{3}{4}\)

Key concepts

Secant Function

The secant function, denoted as \(\sec \theta\), is a fundamental trigonometric function. It is defined as the reciprocal of the cosine function. In mathematical terms, this can be expressed as \(\sec \theta = \frac{1}{\cos \theta}\). This means that whenever you know the value of the secant function, you can immediately find the cosine function by flipping the fraction. In our problem, \(\sec \theta = \frac{5}{3}\), so the cosine function would be \(\cos \theta = \frac{3}{5}\).

Understanding the secant function is crucial, as it helps in finding other trigonometric functions. If you know the secant value, calculating other sine and cosine identities becomes more straightforward. And the position of \(\theta\) within specific quadrants will also influence the sign of these functions. In this case, we are in the fourth quadrant, where cosine is positive.

Sine and Cosine Identities

Sine and cosine identities are at the heart of trigonometry, essential for solving many problems. A particularly important identity is \(\sin^{2} \theta + \cos^{2} \theta = 1\), which links the sine and cosine functions. This identity can be used to find one function if you know the other.

Given our problem, where \(\cos \theta = \frac{3}{5}\), we apply the identity to find \(\sin \theta\). Solving \(\sin^{2} \theta = 1 - \left(\frac{3}{5}\right)^2 = \frac{16}{25}\), you determine \(\sin \theta\). In the fourth quadrant, sine is negative, leading to \(\sin \theta = -\frac{4}{5}\). This demonstrates the identity’s power in solving complex trigonometric expressions.

Tangent and Cotangent

Tangent (\(\tan \theta\)) and cotangent (\(\cot \theta\)) are also crucial trigonometric functions. They can be derived from sine and cosine values. The tangent function is defined as the ratio of sine to cosine: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).

In our exercise, that means \(\tan \theta = \frac{-\frac{4}{5}}{\frac{3}{5}} = -\frac{4}{3}\). Similarly, the cotangent function is the reciprocal of the tangent: \(\cot \theta = \frac{1}{\tan \theta}\), so \(\cot \theta = -\frac{3}{4}\). These functions can be especially helpful when determining angles and calculating trigonometric values in various quadrants.

Cosecant

Cosecant, represented by \(\csc \theta\), is another key trigonometric function tied to sine. It is defined as the reciprocal of the sine function: \(\csc \theta = \frac{1}{\sin \theta}\). This relationship makes it easy to find the cosecant if you know the sine value.

In our problem, \(\sin \theta = -\frac{4}{5}\), so \(\csc \theta = -\frac{5}{4}\). Like secant relates to cosine, cosecant relates to sine. Its understanding is crucial for fully grasping trigonometric functions, especially when working with angles beyond standard positions in a circle. The function's property of being the sine's reciprocal highlights the interconnectedness of trigonometric identities.