right triangle with right angle c. if angle a = pi/6 and a=7 what is the length of side b

We are dealing with a right triangle where:

- \(\angle C = 90^\circ\) (right angle),
- \(\angle A = \frac{\pi}{6}\) (or \(30^\circ\)),
- Side \(a = 7\), which is opposite \(\angle A\),
- We are asked to find the length of side \(b\), which is opposite \(\angle B\).

### Solution

Using trigonometric ratios in a right triangle, we can utilize the **tangent** function, since we know an angle (\(\angle A\)) and the length of the opposite side to that angle (side \(a\)).

#### Tangent formula:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
For \(\angle A = \frac{\pi}{6}\):
\[
\tan\left(\frac{\pi}{6}\right) = \frac{a}{b}
\]
We know \(\tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}}\), and \(a = 7\), so:
\[
\frac{1}{\sqrt{3}} = \frac{7}{b}
\]

Multiplying both sides by \(b\) and solving for \(b\):
\[
b = 7 \sqrt{3}
\]

Thus, the length of side \(b\) is \(7\sqrt{3} \approx 12.12\).

---

Would you like more details, or have any questions?

Here are 5 related questions:
1. How can we calculate the hypotenuse of this triangle?
2. What is the length of side \(c\) (hypotenuse) given angle \(\frac{\pi}{6}\)?
3. How would the result change if \(a\) was doubled?
4. Can we use the sine or cosine function instead of tangent to find side \(b\)?
5. How do we calculate the area of this triangle?

**Tip:** In a right triangle, using the tangent ratio is a quick way to find missing side lengths when you know one

Learn how to calculate the length of side b in a right triangle where angle A is π/6 and side a is 7. This problem involves using trigonometric ratios, specifically the tangent function, to find the missing side. Suitable for high school students (Grades 9-12).

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation