Activity 5. Angle of Inclination. 22 Januaryzoz5 1. Determine the indination of the line. Round your answer to 2 decimal places with points \( A\left(3 ; \frac{7}{2}\right) \) \( B(-8 ;-2) \). 2. Determine the angle of indlination (correct to 1 decimal place) of the straight line through \( (2 ; 1) \notin(-3 ;-9) \) 3. Find the angle \( \theta \) and round off your answer to \( z \)

Sure, let's tackle each of the problems related to the **Angle of Inclination** step by step. ## Understanding Angle of Inclination The **angle of inclination** of a line is the acute angle that the line makes with the positive direction of the x-axis. It is denoted by \( \theta \) and can be calculated using the slope (\( m \)) of the line with the formula: \[ \theta = \arctan(m) \] Where: - \( m \) is the slope of the line. - \( \arctan \) is the inverse tangent function. Let's proceed with each question. --- ## 1. Determining the Inclination of the Line through Points \( A(3, \frac{7}{2}) \) and \( B(-8, -2) \) ### Step 1: Calculate the Slope (\( m \)) of the Line Given two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \), the slope \( m \) is calculated as: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Plugging in the values: \[ m = \frac{-2 - \frac{7}{2}}{-8 - 3} = \frac{-\frac{11}{2}}{-11} = \frac{1}{2} \] So, \( m = 0.5 \). ### Step 2: Calculate the Angle of Inclination (\( \theta \)) \[ \theta = \arctan(0.5) \approx 26.565^\circ \] ### Step 3: Round the Angle to 2 Decimal Places \[ \theta \approx 26.57^\circ \] **Answer:** The angle of inclination is **26.57 degrees**. --- ## 2. Determining the Angle of Inclination of the Straight Line through \( (2, 1) \) and \( (-3, -9) \) ### Step 1: Calculate the Slope (\( m \)) of the Line \[ m = \frac{-9 - 1}{-3 - 2} = \frac{-10}{-5} = 2 \] So, \( m = 2 \). ### Step 2: Calculate the Angle of Inclination (\( \theta \)) \[ \theta = \arctan(2) \approx 63.4349^\circ \] ### Step 3: Round the Angle to 1 Decimal Place \[ \theta \approx 63.4^\circ \] **Answer:** The angle of inclination is **63.4 degrees**. --- ## 3. Finding the Angle \( \theta \) and Rounding Off Your Answer It seems there's incomplete information in **Question 3**. To find the angle \( \theta \), we need either: - Two points defining a line (to calculate the slope), or - The slope of the line directly. If you provide the necessary details or clarify the question, I'd be happy to help you solve it! --- Feel free to reach out if you have more questions or need further clarification on these problems!