Problem 57 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the given values to find the values of the remaining four trigonometric functions of \(\theta\) \(\cos \theta=\frac{5}{13}, \quad \sin \theta=-\frac{12}{13}\)

Step-by-Step Solution

Answer

The remaining four trigonometric functions are \(\tan \theta = -\frac{12}{5}\), \(\sec \theta = \frac{13}{5}\), \(\csc \theta = -\frac{13}{12}\), and \(\cot \theta = -\frac{5}{12}\).

Step 1: Identify the quadrant of the angle

By observing the signs of the cosine and sine functions, we are able to determine the quadrant of \(\theta\). \(\cos \theta=\frac{5}{13}\) is positive and \(\sin \theta=-\frac{12}{13}\) is negative. This combination indicates that the angle \(\theta\) lies in the fourth quadrant.

Step 2: Determine the tangent

Remember that \(\tan \theta=\frac{\sin \theta}{\cos \theta}\). So, using the given values, we get \(\tan \theta=\frac{-12/13}{5/13} = -\frac{12}{5}\).

Step 3: Determine the secant

Recall that the secant is the reciprocal of the cosine, i.e., \(\sec \theta = \frac{1}{\cos \theta}\). Thus, \(\sec \theta = \frac{13}{5}\).

Step 4: Determine the cosecant

The cosecant is the reciprocal of the sine, i.e., \(\csc \theta = \frac{1}{\sin \theta}\). Hence, \(\csc \theta = -\frac{13}{12}\).

Step 5: Determine the cotangent

\(\cot \theta\) is the reciprocal of the tangent. Therefore, \(\cot \theta = -\frac{5}{12}\).

Key Concepts

Cosine and Sine

The cosine and sine functions are fundamental in trigonometry and represent the x and y coordinates, respectively, on the unit circle. Understanding their signs can help determine the quadrant of an angle. Quadrants are divisions of the Cartesian plane:

• In the first quadrant, both cosine and sine are positive.
• In the second, cosine is negative, and sine is positive.
• In the third, both are negative.
• In the fourth, cosine is positive and sine is negative.

Given \(\cos \theta = \frac{5}{13}\) and \(\sin \theta = -\frac{12}{13}\), we find ourselves in the fourth quadrant. These ratios can also be associated with the lengths of sides in a right triangle, where the hypotenuse is 13, the adjacent side is 5, and the opposite side is -12, corresponding to cosine and sine, respectively.

Tangent and Cotangent

Tangent and cotangent are closely related as they represent the slope of a line or the ratio of sine to cosine for tangent:

• \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)
• \(\cot \theta = \frac{\cos \theta}{\sin \theta}\)

Using the given values, \(\tan \theta = \frac{-12/13}{5/13} = -\frac{12}{5}\) and \(\cot \theta = -\frac{5}{12}\). The negative signs are consistent with being in the fourth quadrant, where tangent and cotangent are negative. This reflects the specific angle's behavior in this quadrant, highlighting the slope's downward trend.

Secant and Cosecant

Secant and cosecant are the reciprocal functions for cosine and sine, respectively. They show how these core trigonometric functions have counterparts:

• \(\sec \theta = \frac{1}{\cos \theta}\)
• \(\csc \theta = \frac{1}{\sin \theta}\)

From the cosine \(\cos \theta = \frac{5}{13}\), the secant is \(\sec \theta = \frac{13}{5}\). And from the sine \(\sin \theta = -\frac{12}{13}\), the cosecant is \(\csc \theta = -\frac{13}{12}\). These functions help evaluate angles and determine distances from the origin in different directions, adding depth to the analysis beyond sine and cosine. They also reinforce the concept of reciprocal relationships across different trigonometric functions.