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 $7\sqrt{3} \approx 12.12$.

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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