Problem 6 - Basic Engineering Mathematics

Question

Problem 6. If \(\cos X=\frac{9}{41}\) determine the value of \(\sin X\) and \(\tan X\). Figure \(19.10\) shows a right-angled triangle \(X Y Z\). Fig. \(19.10\) Since \(\cos X=\frac{9}{41}\), then \(X Y=9\) units and \(X Z=41\) units. Using Pythagoras' theorem: \(41^{2}=9^{2}+Y Z^{2}\) from which \(Y Z=\sqrt{41^{2}-9^{2}}=40\) units. Thus \(\sin X=\frac{40}{41} \quad\) and \(\quad \tan X=\frac{40}{9}=4 \frac{4}{9}\)

Step-by-Step Solution

Answer

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

Step 1: Understanding the Right Triangle

In the right triangle, we have \( \cos X = \frac{9}{41} \). This implies that the adjacent side (XY) is 9 units long and the hypotenuse (XZ) is 41 units in length. We need to find the length of the opposite side (YZ) to solve for \( \sin X \) and \( \tan X \).

Step 2: Applying Pythagoras' Theorem

According to Pythagoras' theorem in a right triangle: \( c^2 = a^2 + b^2 \), where \( c \) is the hypotenuse and \( a \) and \( b \) are the other two sides. For triangle XYZ, \( 41^2 = 9^2 + YZ^2 \).

Step 3: Calculating the Opposite Side

First, compute the square of each side: \( 41^2 = 1681 \) and \( 9^2 = 81 \). Using these: \( 1681 = 81 + YZ^2 \). Solving for \( YZ \), we have \( YZ^2 = 1681 - 81 = 1600 \). Thus, \( YZ = \sqrt{1600} = 40 \) units.

Step 4: Finding \( \sin X \)

The sine of angle X is given by \( \sin X = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{40}{41} \).

Step 5: Finding \( \tan X \)

The tangent of angle X is \( \tan X = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{40}{9} \), which is approximately \( 4\frac{4}{9} \).

Key Concepts

Pythagoras' theorem

Pythagoras' theorem is an essential principle in right triangle geometry. It states that in a right-angled triangle, the square of the length of the hypotenuse (\(c\)) is equal to the sum of the squares of the lengths of the other two sides (\(a\) and \(b\)). This relationship can be expressed as \[ c^2 = a^2 + b^2 \].In our exercise, the triangle in question has been described in terms of cosine and side lengths. The known measures are the lengths of the adjacent side and the hypotenuse. By Pythagoras' theorem, we can find the missing side, the opposite side. Substituting the given values, \( 41^2 = 9^2 + YZ^2 \), where 41 is the hypotenuse and 9 is the adjacent side. Solving this equation helps us find \( YZ \), the opposite side, completing the understanding of the triangle's dimensions.Remember:

• The hypotenuse is the longest side of a right triangle.
• Use Pythagoras' theorem only in right-angled triangles.

Understanding this theorem is crucial for solving various trigonometric problems effectively and accurately.

right triangle geometry

Right triangle geometry involves studying triangles that have one angle equal to 90 degrees. These triangles have particular properties that allow us to explore and solve using specific rules and relationships. The right triangle consists of three main elements:

• The hypotenuse, which is the side opposite the right angle and the triangle's longest side.
• The legs or catheti, which are the two sides that form the right angle.
• The angles, specifically the right angle and the two acute angles.

In our exercise, understanding the right triangle's structure allows us to utilize trigonometric identities and theorems effectively. By assigning names to each part of the triangle (like \(XY, XZ,\) and \(YZ\) in our problem), it becomes easier to apply the rules like Pythagoras' theorem. Therefore, approaching the problem step-by-step, without rushing, ensures clarity and accuracy in solving for unknowns like the side lengths or angles.

sine and cosine relationships

Sine and cosine are fundamental trigonometric functions that relate the angles and sides of a right triangle. These functions provide the backbone for solving various geometric and applied problems.- **Cosine Function (cos)**: - Cosine of an angle \(X\) in a right triangle is the ratio of the length of the adjacent side to the hypotenuse. In our exercise, \( \cos X = \frac{9}{41} \) helped determine the lengths of the sides initially.- **Sine Function (sin)**: - Sine of an angle \(X\) is the ratio of the length of the opposite side to the hypotenuse. After finding the opposite side as 40 units, we find \( \sin X = \frac{40}{41} \).These relationships help build a bridge between angles and side lengths:

• Sine tells us how tall the triangle is relative to its hypotenuse.
• Cosine tells us how far the triangle is along the hypotenuse direction.

Understanding these functions and their ratios opens up the pathway to understanding angles, solving for missing parts of triangles, and even modeling real-world problems using trigonometry.