Problem 48 - Precalculus Mathematics for Calculus

Question

Find the values of the trigonometric functions of \(\theta\) from the information given. $$\sec \theta=5, \quad \sin \theta<0$$

Step-by-Step Solution

Answer

In quadrant IV, \( \cos \theta = \frac{1}{5} \), \( \sin \theta = -\frac{2\sqrt{6}}{5} \), and \( \tan \theta = -2\sqrt{6} \).

Step 1: Relationship between secant and cosine

The secant of an angle is the reciprocal of the cosine of the angle. So, if \( \sec \theta = 5 \), then \( \cos \theta = \frac{1}{5} \).

Step 2: Determine the sign of cosine and sine

Since \( \sec \theta = 5 \) is positive, \( \cos \theta \) is positive. Given \( \sin \theta < 0 \), this information tells us that \( \theta \) is in the fourth quadrant, where cosine is positive and sine is negative.

Step 3: Use the Pythagorean identity

The Pythagorean identity states \( \sin^2 \theta + \cos^2 \theta = 1 \). Substituting \( \cos \theta = \frac{1}{5} \), we have \[ \sin^2 \theta + \left(\frac{1}{5}\right)^2 = 1 \] \[ \sin^2 \theta + \frac{1}{25} = 1 \].

Step 4: Solve for \( \sin \theta \)

Rearranging the equation from Step 3, we get \( \sin^2 \theta = 1 - \frac{1}{25} \), which simplifies to \[ \sin^2 \theta = \frac{25}{25} - \frac{1}{25} = \frac{24}{25} \]. \[\sin \theta = -\sqrt{\frac{24}{25}} = -\frac{\sqrt{24}}{5} = -\frac{2\sqrt{6}}{5}\] since we know \( \sin \theta < 0 \) in the fourth quadrant.

Step 5: Calculate \( \tan \theta \)

Using the identity \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), substitute \( \sin \theta = -\frac{2\sqrt{6}}{5} \) and \( \cos \theta = \frac{1}{5} \), giving \[ \tan \theta = \frac{-\frac{2\sqrt{6}}{5}}{\frac{1}{5}} = -2\sqrt{6} \].

Key Concepts

Secant Function

The secant function is a fundamental trigonometric function that is the reciprocal of the cosine function. In simpler terms, secant (\( \sec \theta \)) can be visualized as \( \sec \theta = \frac{1}{\cos \theta} \).
When you have \( \sec \theta = 5 \), it means that the cosine of the angle, \( \cos \theta \), equals \( \frac{1}{5} \).
Key points to remember about secant:

• Secant is undefined for angles where cosine is zero since division by zero is undefined.
• In terms of the unit circle, the secant function represents the reciprocal of the x-coordinate of a point on the unit circle.

Recognizing the relationship between secant and cosine is important since it can help you determine other trigonometric values based on given information.

Pythagorean Identity

The Pythagorean identity is a cornerstone in trigonometry that relates the square of sine and cosine functions. Specifically, it is expressed as \( \sin^2 \theta + \cos^2 \theta = 1 \). This identity holds true for every angle \( \theta \).
It directly arises from the Pythagorean theorem applied to a unit circle, where a right triangle is formed with the hypotenuse as the radius of the circle, measuring 1 unit.
In our exercise, knowing that \( \cos \theta = \frac{1}{5} \), we can substitute into the identity:

• \( \sin^2 \theta + \left(\frac{1}{5}\right)^2 = 1 \)
• This simplifies to \( \sin^2 \theta = 1 - \frac{1}{25} = \frac{24}{25} \).

To find \( \sin \theta \), we take the square root, giving \( \sin \theta = -\frac{2\sqrt{6}}{5} \), choosing the negative root due to \( \sin \theta < 0 \). Understanding this identity enables solving problems that call for determining sine or cosine given partial information.

Sine and Cosine Relationship

The sine and cosine functions have an intrinsic relationship that is crucial to understanding circle-based trigonometry. Both functions can be visualized on the unit circle, with sine representing the y-coordinate and cosine the x-coordinate of a point corresponding to an angle.
In trigonometry, this relationship is foundational,

• Expressed through their complementary angles, where \( \sin(90^\circ - \theta) = \cos \theta \) and \( \cos(90^\circ - \theta) = \sin \theta \).
• Both sine and cosine values are also influenced by which quadrant an angle lies in, supporting the nature of their signs. In the fourth quadrant, sine is negative, and cosine is positive.

For the problem at hand, we calculated \( \sin \theta \) using the Pythagorean identity and affirmed the sign based on the fourth-quadrant rule. Understanding how sine and cosine relate provides a broader comprehension of how to apply these functions across various trigonometric problems.