Problem 19 - Precalculus with Calculus Previews

Question

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

Step-by-Step Solution

Answer

\( \sin x = \frac{2}{\sqrt{5}} \), \( \cos x = -\frac{1}{\sqrt{5}} \), \( \sec x = -\sqrt{5} \), \( \csc x = \frac{\sqrt{5}}{2} \), \( \cot x = -\frac{1}{2} \).

Step 1: Determine the Quadrant

Since \( \pi/2 < x < \pi \), angle \( x \) is in the second quadrant. In this quadrant, the sine function is positive while the cosine and tangent functions are negative.

Step 2: Recall Trigonometric Identities

Recall that the basic trigonometric identities are \( \tan x = \frac{\sin x}{\cos x} \), \( \sin^2 x + \cos^2 x = 1 \), \( \sec x = \frac{1}{\cos x} \), \( \csc x = \frac{1}{\sin x} \), and \( \cot x = \frac{1}{\tan x} \).

Step 3: Use the Given \( \tan x \)

Given \( \tan x = -2 \), we know that \( \frac{\sin x}{\cos x} = -2 \). Hence, \( \sin x = -2\cos x \).

Step 4: Apply Pythagorean Identity

Substitute \( \sin x = -2\cos x \) into the Pythagorean identity, \( \sin^2 x + \cos^2 x = 1 \). This results in \( (-2\cos x)^2 + \cos^2 x = 1 \).

Step 5: Simplify and Solve for \( \cos x \)

\((-2\cos x)^2 = 4\cos^2 x\), so the equation becomes \(4\cos^2 x + \cos^2 x = 1\). Combine terms to get \(5\cos^2 x = 1\), thus \(\cos^2 x = \frac{1}{5}\). Take the square root: \(\cos x = -\frac{1}{\sqrt{5}}\) because \(\cos x\) is negative in the second quadrant.

Step 6: Find \( \sin x \) from \( \cos x \)

Substitute back \( \cos x = -\frac{1}{\sqrt{5}} \) into \( \sin x = -2\cos x \) to find \( \sin x = -2\left(-\frac{1}{\sqrt{5}}\right) = \frac{2}{\sqrt{5}}\).

Step 7: Calculate the Remaining Trigonometric Functions

Now that \( \cos x = -\frac{1}{\sqrt{5}} \) and \( \sin x = \frac{2}{\sqrt{5}} \), calculate:- \( \sec x = \frac{1}{\cos x} = -\sqrt{5} \)- \( \csc x = \frac{1}{\sin x} = \frac{\sqrt{5}}{2} \)- \( \cot x = \frac{1}{\tan x} = -\frac{1}{2} \)

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical functions that relate angles of a triangle to the ratios of its sides. These functions are fundamental in trigonometry, and they include:

• Sine (\( \sin \)
• Cosine (\( \cos \)
• Tangent (\( \tan \)
• Cosecant (\( \csc \)
• Secant (\( \sec \)
• Cotangent (\( \cot \)

Tangent is the ratio of sine over cosine, written as \( \tan x = \frac{\sin x}{\cos x} \).Given in the exercise, we know \( \tan x = -2 \).This tells us essential information about our angle, specifically that the sine value is twice the magnitude of the cosine (with opposite signs). Understanding these relationships helps solve for unknown trigonometric values.

Pythagorean Identity

One of the most important tools in trigonometry is the Pythagorean identity, which is key to relating different trigonometric functions:\[\sin^2 x + \cos^2 x = 1\]This identity comes from the Pythagorean Theorem applied to the unit circle, and it implies that for any angle x, the sum of the squares of the sine and cosine is always equal to one. In the given exercise, this identity helps us find unknown trigonometric functions when only one is known. By substituting known values like \( \sin x = -2\cos x \) into the identity, we find values for \( \cos x \) and subsequently all other functions.Correctly applying the Pythagorean identity facilitates solving for complex problems with relative ease.

Unit Circle

The unit circle is a circle of radius one, centered at the origin of a coordinate plane. It's an essential concept in trigonometry because it provides a geometric interpretation of trigonometric functions.On the unit circle:

• The cosine of an angle is the x-coordinate of the corresponding point.
• The sine of an angle is the y-coordinate.
• The tangent is the ratio of the sine to the cosine.

The unit circle also helps visualize positive and negative signs of functions in different quadrants.In our case, when determining trigonometric values for an angle in the second quadrant, the unit circle reminds us that \( \cos x \) is negative and \( \sin x \) is positive.

Second Quadrant

Understanding quadrants in the coordinate plane is crucial for mastering trigonometric identities. The second quadrant is defined where:

• x-values are negative
• y-values are positive

Thus, in the second quadrant:

• Cosine \( \cos x \) is negative
• Sine \( \sin x \) is positive
• Tangent \( \tan x \) is negative due to \( \sin x \) being divided by a negative \( \cos x \)

In the exercise, knowing the angle was in the second quadrant, we determined \( \sin x = \frac{2}{\sqrt{5}} \) and \( \cos x = -\frac{1}{\sqrt{5}} \), which was consistent with the quadrant's sign rules.Recognizing these properties allows you to correctly predict the signs and sometimes the values of various trigonometric functions just based on their quadrant.