Problem 22 - Precalculus with Calculus Previews

Question

Use the given information to find the values of the remaining five trigonometric functions. $$ \sec x=-5, \pi / 2

Step-by-Step Solution

Answer

\(\cos x = -\frac{1}{5}, \sin x = \frac{\sqrt{24}}{5}\) \(\tan x = -\sqrt{24}, \csc x = \frac{5\sqrt{24}}{24}\) \(\cot x = -\frac{\sqrt{24}}{24}\)

Step 1: Determine the quadrant

Given that \(-\frac{\pi}{2} < x < \pi\), the angle \(x\) is located in the second quadrant. In the second quadrant, the sine function is positive, and the cosine function is negative.

Step 2: Find the cosine of \(x\)

The secant function is the reciprocal of the cosine function, so \(\sec x = -5\) implies \(\cos x = -\frac{1}{5}\).

Step 3: Determine the sine of \(x\)

Use the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\). Substitute \(\cos x = -\frac{1}{5}\) into the formula: \[\sin^2 x + \left(-\frac{1}{5}\right)^2 = 1\]\[\sin^2 x + \frac{1}{25} = 1\]\[\sin^2 x = 1 - \frac{1}{25} = \frac{24}{25}\]\[\sin x = \sqrt{\frac{24}{25}}\]Since \(x\) is in the second quadrant, \(\sin x\) is positive. Therefore, \(\sin x = \frac{\sqrt{24}}{5}\).

Step 4: Calculate the tangent of \(x\)

Tangent is calculated as \(\tan x = \frac{\sin x}{\cos x}\) so substituting the values:\[\tan x = \frac{\frac{\sqrt{24}}{5}}{-\frac{1}{5}} = -\sqrt{24}\]

Step 5: Find the cosecant and cotangent of \(x\)

Cosecant is the reciprocal of sine: \[\csc x = \frac{1}{\sin x} = \frac{5}{\sqrt{24}}\]Rationalizing gives us: \[\csc x = \frac{5\sqrt{24}}{24}\] Cotangent is the reciprocal of tangent:\[\cot x = \frac{1}{\tan x} = -\frac{1}{\sqrt{24}}\] Rationalizing gives us: \[\cot x = -\frac{\sqrt{24}}{24}\]

Key Concepts

Secant Function

The secant function, denoted as \( \sec x \), is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the cosine function. In simpler terms, if \( \cos x = a \), then \( \sec x = \frac{1}{a} \). Typical values for \( \sec x \) range from \(-\infty\) to \(-1\) or from \(1\) to \(\infty\).
Secant can also be understood geometrically on the unit circle: while the cosine corresponds to the adjacent side of the triangle drawn from the origin to the circle, the secant represents the length of the hypotenuse extending beyond this adjacent line segment. This relationship makes secant a useful function in understanding periodic phenomena and in solving geometric problems.

Pythagorean Identity

The Pythagorean identity is a cornerstone in trigonometry. It states that for any angle \(x\), the square of the sine of \(x\) plus the square of the cosine of \(x\) equals one: \( \sin^2 x + \cos^2 x = 1 \). This identity is tremendously useful in expressing one trigonometric function in terms of another.
In the exercise, we use the Pythagorean identity to find \( \sin x \) after determining \( \cos x \) from \( \sec x \).

• Given \( \cos x = -\frac{1}{5} \)
• Substitute into the identity to find \( \sin x \)
• \( \sin^2 x + \left(-\frac{1}{5}\right)^2 = 1 \)
• Simplifies to \( \sin^2 x = \frac{24}{25} \)

This shows how interconnected trig functions can express the values of other trigonometric functions.

Second Quadrant

The second quadrant, in the context of the unit circle and trigonometry, is where the angle \(x\) lies between \( \frac{\pi}{2} \) and \( \pi \) radians. In this quadrant, certain characteristics of trigonometric functions are notable. Here, sine values are positive, and cosine values are negative.
Understanding which quadrant an angle is in helps predict the sign of trigonometric functions without direct computation. For example:

• Since \( \sec x = -5 \), a negative value, indicates \( \cos x = -\frac{1}{5} \) is indeed negative
• Using this knowledge, we confirm that \( \sin x \) should be positive in the second quadrant
• This affects the signs of other trigonometric components, such as \( \tan x \), which would be negative due to the ratio \( \frac{\sin x}{\cos x} \)

Identifying the quadrant simplifies solving and verifying trigonometric expressions.

Sec x

\( \sec x \) is a critical trigonometric function that equals the reciprocal of the cosine function, or \( \sec x = \frac{1}{\cos x} \). In our specific exercise, \( \sec x = -5 \), it allows us to directly find \( \cos x = -\frac{1}{5} \).
Understanding how \( \sec x \) operates can be vital in solving trigonometric equations as it provides a straightforward method to switch between secant and cosine.

• If you know \( \sec x \), as in our problem at \( -5 \), simply take the reciprocal to find \( \cos x \)
• This straightforward relationship helps with solving further trigonometric properties, like double or half-angle identities

By mastering the interplay between \( \sec x \) and \( \cos x \), complex problems become much easier to tackle.

Reciprocal Identities

Trigonometric reciprocal identities are formulas involving the reciprocals of the basic trigonometric functions. Each of the main trigonometric functions (sine, cosine, tangent) has a reciprocal: cosecant (\( \csc x \)), secant (\( \sec x \)), and cotangent (\( \cot x \)).
These identities are:

• \( \csc x = \frac{1}{\sin x} \)
• \( \sec x = \frac{1}{\cos x} \)
• \( \cot x = \frac{1}{\tan x} \)

Reciprocal identities are essential because they allow us to interchange between different forms of trigonometric expressions, making it easier to solve complex equations. For example:

• In the exercise, once we find \( \sin x \), we can immediately determine \( \csc x \) by computing its reciprocal
• Similarly, knowing \( \tan x \) allows easy calculation of \( \cot x \)

These identities develop a deeper understanding of trigonometric circles and oscillatory motion.