Problem 3 - Basic Engineering Mathematics

Question

If \(\cos X=\frac{9}{41}\) determine the value of the other five trigonometry ratios.

Step-by-Step Solution

Answer

\( \sin X = \frac{40}{41} \), \( \tan X = \frac{40}{9} \), \( \csc X = \frac{41}{40} \), \( \sec X = \frac{41}{9} \), \( \cot X = \frac{9}{40} \).

Step 1: Identify Given Information

We are given that \( \cos X = \frac{9}{41} \). This means the adjacent side over the hypotenuse in a right-angled triangle is given by this ratio.

Step 2: Employ Pythagorean Identity

Recall the identity \( \sin^2 X + \cos^2 X = 1 \). Substitute \( \cos X = \frac{9}{41} \) to find \( \sin X \).\[ \sin^2 X = 1 - \left(\frac{9}{41}\right)^2 \]\[ \sin^2 X = 1 - \frac{81}{1681} \]\[ \sin^2 X = \frac{1600}{1681} \]Taking the square root gives:\[ \sin X = \frac{40}{41} \]Assuming \( X \) is in the first quadrant, \( \sin X \) is positive.

Step 3: Determine \( \tan X \)

\( \tan X \) is the ratio \( \frac{\sin X}{\cos X} \).\[ \tan X = \frac{\frac{40}{41}}{\frac{9}{41}} = \frac{40}{9} \]

Step 4: Find \( \csc X \)

The cosecant function is the reciprocal of \( \sin X \).\[ \csc X = \frac{1}{\sin X} = \frac{41}{40} \]

Step 5: Calculate \( \sec X \)

The secant function is the reciprocal of \( \cos X \).\[ \sec X = \frac{1}{\cos X} = \frac{41}{9} \]

Step 6: Get \( \cot X \)

The cotangent is the reciprocal of \( \tan X \).\[ \cot X = \frac{1}{\tan X} = \frac{9}{40} \]

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a fundamental concept in trigonometry that connects the sine and cosine functions. It's based on the Pythagorean theorem, which relates the sides of a right-angled triangle.
To understand this identity better, recall the expression:

• \( \sin^2 X + \cos^2 X = 1 \)

This equation tells us that if we square both the sine and cosine of an angle and add them together, the result is always one. This identity is extremely useful in finding unknown trigonometric values, as it helps in transforming or simplifying expressions.

In the given problem, we started with \( \cos X = \frac{9}{41} \), and used the Pythagorean Identity to find \( \sin X \). Knowing either cosine or sine, you can always use this identity to compute the other value. Just substitute the given value into the identity and solve for the unknown.

Reciprocal Trigonometric Functions

In trigonometry, reciprocal functions are very important, particularly when dealing with trigonometric identities. Each primary trigonometric function has a reciprocal function:

• The reciprocal of sine (\( \sin X \)) is cosecant (\( \csc X \)).
• For cosine (\( \cos X \)), it's secant (\( \sec X \)).
• The reciprocal of tangent (\( \tan X \)) is cotangent (\( \cot X \)).

Using reciprocal functions allows us to solve trigonometry problems more efficiently. For instance, in our exercise, knowing \( \sin X \) helps us find \( \csc X \) by simply taking its reciprocal:
\[ \csc X = \frac{1}{\sin X} \]
Similarly, for \( \cos X \) and \( \sec X \), the equation is:
\[ \sec X = \frac{1}{\cos X} \]
These conversions help immensely in trigonometric calculations by making it easier to work through without complex conversions or adjustments.

First Quadrant

The first quadrant is a specific region within the Cartesian coordinate system, and it's very important in trigonometry for determining the signs of trigonometric functions.
In the first quadrant, angles range from 0 to 90 degrees (or 0 to \( \frac{\pi}{2} \) radians). Here, both x and y coordinates are positive, which means:

• \( \sin X \) is positive.
• \( \cos X \) is also positive.
• Thus, \( \tan X \), which is calculated as \( \frac{\sin X}{\cos X} \), is positive too.

Understanding this is crucial when solving trigonometric problems, as knowing that an angle is in the first quadrant simplifies determining whether trigonometric values are positive or negative. This was used in the solution to assume \( \sin X \) is positive, reflecting the convention that angles situated in this quadrant result in positive values for their sine, cosine, and tangent functions.