https://file.aiquickdraw.com/mathbot/file/17255977042152a0bj9pw.jpg

Find the length of the opposite side, b.

Write each line of working as an equation.

This is a right-angled triangle. To find the length of the opposite side (denoted as \( b \)), we'll use **trigonometry**. Based on the diagram, we can use the sine function, as it relates the opposite side, hypotenuse, and angle \(\theta\).

Here are the steps:

1. **Identify the sides and angle:**
   - Opposite side to \(\theta\) is the side \( BC \), which we are calling \( b \).
   - The hypotenuse is side \( AB \), which is 5 units.

2. **Apply the sine function:**
   - The sine of an angle is the ratio of the opposite side to the hypotenuse:
   \[
   \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{b}{5}
   \]
   
3. **Solve for \( b \):**
   \[
   b = 5 \times \sin(\theta)
   \]

4. **Determine \(\theta\):**
   Since \( \triangle ABC \) is a right triangle and \( \angle ACB = 90^\circ \), we can use the tangent function to find \( \theta \). The tangent function is the ratio of the opposite side to the adjacent side:
   \[
   \tan(\theta) = \frac{3}{4}
   \]
   
   Using inverse tangent:
   \[
   \theta = \tan^{-1}\left(\frac{3}{4}\right)
   \]

5. **Calculate \( \theta \) and then \( b \):**
   Let's now calculate \( \theta \) and the final length of the opposite side.

I will compute that next.The angle \( \theta \) is approximately \( 36.87^\circ \).

Using this angle, the length of the opposite side \( b \) is calculated as 3 units.

This matches the given value for side \( BC \), confirming that the calculation is correct.

Would you like more details or have any questions? Here are some related questions for further understanding:

1. What is the relationship between sine, cosine, and tangent in a right triangle?
2. How can the cosine function be used to solve for an adjacent side?
3. What is the Pythagorean theorem, and how can it verify the side lengths of a triangle?
4. How can you calculate the other angles in the triangle using trigonometric ratios?
5. How does the inverse sine function work in solving for angles?

**Tip:** Always make sure your calculator is set to the correct mode (degrees or radians) when performing trigonometric calculations.

Learn how to find the length of the opposite side in a right triangle using sine and tangent functions. This trigonometry problem involves calculating θ and applying trigonometric ratios. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation