Problem 24 - Precalculus with Calculus Previews

Question

Use the given information to find the values of the remaining five trigonometric functions. $$ \cos x=-1 / \sqrt{5}, \quad \pi

Step-by-Step Solution

Answer

The other trigonometric functions are \( \sin x = -\frac{2}{\sqrt{5}} \), \( \tan x = 2 \), \( \cot x = \frac{1}{2} \), \( \sec x = -\sqrt{5} \), \( \csc x = -\frac{\sqrt{5}}{2} \).

Step 1: Identify the Quadrant

Given that \( \pi < x < 3\pi / 2 \), \( x \) is in the third quadrant. In the third quadrant, both sine and cosine values are negative, while tangent and cotangent are positive.

Step 2: Use the Pythagorean Identity

The Pythagorean identity is \( \sin^2 x + \cos^2 x = 1 \). We have \( \cos x = -\frac{1}{\sqrt{5}} \), so substitute this into the identity:\[\sin^2 x + \left(-\frac{1}{\sqrt{5}}\right)^2 = 1 \\sin^2 x + \frac{1}{5} = 1\] This simplifies to:\[\sin^2 x = 1 - \frac{1}{5} = \frac{4}{5}\] Taking the square root of both sides, \( \sin x = -\frac{2}{\sqrt{5}} \) since \( \sin x < 0 \) in the third quadrant.

Step 3: Calculate Tangent

Now that we know \( \sin x \) and \( \cos x \), we can find \( \tan x \) using the formula \( \tan x = \frac{\sin x}{\cos x} \):\[\tan x = \frac{-\frac{2}{\sqrt{5}}}{-\frac{1}{\sqrt{5}}} = 2\]

Step 4: Calculate Cotangent

The cotangent is the reciprocal of tangent. Therefore, \( \cot x = \frac{1}{\tan x} \):\[\cot x = \frac{1}{2}\]

Step 5: Calculate Secant

Secant is the reciprocal of cosine, so \( \sec x = \frac{1}{\cos x} \):\[\sec x = -\sqrt{5}\]

Step 6: Calculate Cosecant

Cosecant is the reciprocal of sine, so \( \csc x = \frac{1}{\sin x} \):\[\csc x = -\frac{\sqrt{5}}{2}\]

Key Concepts

Understanding Cosine

Cosine is one of the main trigonometric functions and it's usually denoted as \( \cos \). It relates to the adjacent side and hypotenuse of a right-angled triangle. In simple terms, cosine is the ratio of the length of the adjacent side to the length of the hypotenuse. When you deal with angles on the unit circle, which has a radius of 1, cosine gives you the x-coordinate of the angle.
For example:

• If you have an angle \( x \) and \( \cos x = -\frac{1}{\sqrt{5}} \), it tells you the direction on the x-axis from the origin to that angle point.
• Cosine values can be positive or negative depending on the angle's quadrant. In this exercise, since we are in the third quadrant, \( \cos \) is negative.

Cosine is an essential part of finding the other trigonometric functions using identities and other functions.

Diving into Sine

Sine is a close relative of cosine in the set of trigonometric functions and is denoted as \( \sin \). It represents the ratio of the opposite side over the hypotenuse in a right-angled triangle. On the unit circle, sine gives us the y-coordinate of an angle. In this particular problem:

• We have \( \cos x = -\frac{1}{\sqrt{5}} \), so we use the Pythagorean identity to find sine.
• The Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \) helps us calculate \( \sin x \) knowing \( \cos x \).
• Since we are in the third quadrant, where sine is negative, we find \( \sin x = -\frac{2}{\sqrt{5}} \).

Sine values will also vary depending on the angle's quadrant, with positive values often found in quadrants I and II.

Understanding Tangent

Tangent, another fundamental trigonometric function, delivers the ratio of sine over cosine, \( \tan = \frac{\sin}{\cos} \). It helps in understanding the slope of the angle in a coordinate system, essentially describing the angle's steepness or flatness.

• When you have both \( \sin x \) and \( \cos x \) values, you can simply divide them \( \tan x = \frac{-\frac{2}{\sqrt{5}}}{-\frac{1}{\sqrt{5}}} = 2 \).
• Tangent is positive in the third quadrant, aligning with the inclination that affects slopes of downward directions in mathematics.

Remember, knowing \( \tan x \) makes it easier to understand angles and their tangent's sign depends on the angle's quadrant position.

The Pythagorean Identity Insight

The Pythagorean identity is a cornerstone in trigonometry. It offers a way to relate sine and cosine in a simple equation: \( \sin^2 x + \cos^2 x = 1 \). This identity is handy to calculate unknown trigonometric values.

• You can rearrange this identity to solve for \( \sin \) or \( \cos \) when you have one value. This gives insights into the relationships between these trigonometric functions.
• In our task, with \( \cos x = -\frac{1}{\sqrt{5}} \), we used it to find \( \sin x \), confirming that \( \sin^2 x = \frac{4}{5} \).
• It's essential to note the negative and positive nature of sine and cosine depending on the quadrant.

Understanding the Pythagorean identity makes it easier to manipulate and solve trigonometric functions, aiding in tasks like calculating angles and triangulating positions.