If Δ ABC is right angled at B, AB = 12 cm and ∠CAB = 60°, determine the length of BC.
12√3 cm
Solving for Side Length in a Right-Angled Triangle using Trigonometry
The problem asks us to find the length of the side BC in a right-angled triangle ABC, where the right angle is at B. We are given the length of side AB and the measure of angle CAB.
Understanding the Given Information
- Triangle ABC is right-angled at B, which means $\ang$ABC = 90°.
- The length of side AB is 12 cm. This side is adjacent to angle CAB.
- The measure of angle CAB ($\ang$CAB) is 60°.
- We need to determine the length of side BC, which is opposite to angle CAB.
Applying Trigonometry to the Right Triangle
In a right-angled triangle, we can use trigonometric ratios (sine, cosine, tangent) to relate the angles to the side lengths. For angle CAB (60°):
- The side opposite to $\ang$CAB is BC.
- The side adjacent to $\ang$CAB is AB.
- The hypotenuse is AC (opposite the right angle).
We have the adjacent side (AB) and need to find the opposite side (BC) with respect to the given angle (60°). The trigonometric ratio that connects the opposite side and the adjacent side is the tangent function:
$\text{tan}(\theta) = \frac{\text{Opposite side}}{\text{Adjacent side}}$
Calculating the Length of BC
Using the tangent function with $\theta = \ang$CAB = 60°:
$\text{tan}(\ang\text{CAB}) = \frac{\text{BC}}{\text{AB}}$
Substitute the given values:
$\text{tan}(60°) = \frac{\text{BC}}{12}$
We know that the value of $\text{tan}(60°)$ is $\sqrt{3}$.
So, the equation becomes:
$\sqrt{3} = \frac{\text{BC}}{12}$
To find BC, we multiply both sides of the equation by 12:
$\text{BC} = 12 \times \sqrt{3}$
$\text{BC} = 12\sqrt{3}$ cm
Therefore, the length of side BC is $12\sqrt{3}$ cm.
Revision Table: Key Trigonometric Values for Common Angles
| Angle ($\theta$) | sin($\theta$) | cos($\theta$) | tan($\theta$) |
|---|---|---|---|
| 30° | $\frac{1}{2}$ | $\frac{\sqrt{3}}{2}$ | $\frac{1}{\sqrt{3}}$ |
| 45° | $\frac{1}{\sqrt{2}}$ | $\frac{1}{\sqrt{2}}$ | 1 |
| 60° | $\frac{\sqrt{3}}{2}$ | $\frac{1}{2}$ | $\sqrt{3}$ |
| 90° | 1 | 0 | Undefined |
Additional Information: Soh Cah Toa
The relationships between the angles and sides in a right-angled triangle are often remembered using the acronym SOH CAH TOA:
- SOH: Sine = Opposite / Hypotenuse
- CAH: Cosine = Adjacent / Hypotenuse
- TOA: Tangent = Opposite / Adjacent
In this problem, we used TOA because we had the adjacent side and needed to find the opposite side, given the angle.
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