Problem 2 - Basic Engineering Mathematics

Question

Determine all the angles between \(0^{\circ}\) and \(360^{\circ}\) whose tangent is \(1.7629\) A tangent is positive in the first and third quadrants \(-\) see Fig. \(20.8\). From Fig. 20.9, \(\theta=\tan ^{-1} 1.7629=60.44^{\circ}\) Fig. \(\mathbf{2 0 . 8}\) Fig. 20.9 Measured from \(0^{\circ}\), the two angles between \(0^{\circ}\) and \(360^{\circ}\) whose tangent is \(1.7629\) are \(60.44^{\circ}\) and \(180^{\circ}+60.44^{\circ}\), i.e. \(240.44^{\circ}\)

Step-by-Step Solution

Answer

The angles are 60.44° and 240.44°.

Step 1: Understand tangent function properties

The tangent function is positive in the first quadrant (0° to 90°) and the third quadrant (180° to 270°). We will use this property to find all angles whose tangent is 1.7629.

Step 2: Find angle in the first quadrant

Calculate the angle in the first quadrant using the inverse tangent function. This is done by finding \( \theta = \tan^{-1}(1.7629) \). Thus, \( \theta = 60.44^{\circ} \).

Step 3: Determine angle in the third quadrant

Since the tangent function is also positive in the third quadrant, find the angle by adding 180° to the first quadrant angle: \( 180^{\circ} + 60.44^{\circ} = 240.44^{\circ} \).

Step 4: List all possible angles

Combine the results from Steps 2 and 3. The angles between 0° and 360° whose tangent is 1.7629 are 60.44° and 240.44°.

Key Concepts

Tangent Function

The tangent function, often represented as \( \tan(\theta) \), is a fundamental concept in trigonometry. It relates the angle \( \theta \) of a right triangle to the ratio of the opposite side over the adjacent side. In the standard unit circle, this becomes the ratio of the y-coordinate to the x-coordinate at a given angle.

Key properties of the tangent function include:

• It has a periodicity of 180°, which means \( \tan(\theta) = \tan(\theta + 180°) \).
• The function is undefined at \( 90° + k \cdot 180° \) (where \( k \) is an integer), since the denominator approaches zero at these points.
• Its range is all real numbers, but it repeats every 180°.

In the context of this exercise, the tangent function is particularly useful because we are looking for angles where the tangent has a specific positive value (1.7629).

Remember, this function is positive in both the first quadrant (0° to 90°) and the third quadrant (180° to 270°). These quadrant properties help us simplify finding the solutions in tasks involving the tangent function.

Inverse Trigonometric Functions

Inverse trigonometric functions are tools that allow us to find angles when given a specific trigonometric y-value. For the tangent function, its inverse is denoted as \( \tan^{-1}(x) \), and it outputs an angle whose tangent is \( x \).

Let's dive into the main characteristics of \( \tan^{-1}(x) \):

• It is also known as the arctangent function.
• Its range is \((-90°, 90°)\), which corresponds to the first and fourth quadrants, but in a broader sense encompasses angles from -\(\frac{\pi}{2} \) to \(\frac{\pi}{2}\) radians.
• It is useful for solving equations where you need to find an angle given its tangent ratio.

In the provided exercise, we used \( \tan^{-1} \) to determine the first angle \( (60.44°) \) in the first quadrant where the tangent precisely equals \( 1.7629 \). Understanding how to apply inverse functions can streamline solving many trigonometric equation problems.

Quadrants of a Circle

The quadrants of a circle are divisions of the plane around the circle, based on the Cartesian coordinate system. They are crucial for understanding trigonometric functions, which can be positive or negative depending on the quadrant.

Here's a quick overview:

• **First Quadrant**: Both sine and cosine are positive, and hence the tangent is positive. This covers angles from 0° to 90°.
• **Second Quadrant**: Sine is positive, cosine is negative, making the tangent negative. This covers angles from 90° to 180°.
• **Third Quadrant**: Both sine and cosine are negative, thus the tangent is positive. This ranges from 180° to 270°.
• **Fourth Quadrant**: Sine is negative, cosine is positive, so the tangent remains negative. Angles span from 270° to 360°.

In the exercise context, the importance lies in knowing where the tangent function acquires its positive attitudes. Using this, we conclude the required angles (60.44° and 240.44°) by utilizing the first and third quadrants effectively.