Trigonometry: Finding Sine from Sine/Cosine Ratio

We are given the ratio of the sine of an acute angle to its cosine is 5 : 12.

This can be written as:

$ \frac{\sin(\theta)}{\cos(\theta)} = \frac{5}{12} $

We know that the tangent of an angle is defined as the ratio of its sine to its cosine:

$ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} $

Therefore, we have:

$ \tan(\theta) = \frac{5}{12} $

Constructing a Right-Angled Triangle

For an acute angle $\theta$ in a right-angled triangle:

$ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} $

Given $\tan(\theta) = \frac{5}{12}$, we can represent the sides of the triangle. Let the length of the opposite side be $5k$ and the length of the adjacent side be $12k$, where $k$ is a positive constant.

Calculating the Hypotenuse

Using the Pythagorean theorem ($a^2 + b^2 = c^2$), where $a$ and $b$ are the lengths of the two shorter sides (opposite and adjacent) and $c$ is the length of the hypotenuse:

$ (\text{Opposite})^2 + (\text{Adjacent})^2 = (\text{Hypotenuse})^2 $

$ (5k)^2 + (12k)^2 = (\text{Hypotenuse})^2 $

$ 25k^2 + 144k^2 = (\text{Hypotenuse})^2 $

$ 169k^2 = (\text{Hypotenuse})^2 $

Taking the square root of both sides:

$ \text{Hypotenuse} = \sqrt{169k^2} = 13k $

Determining the Sine Value

The sine of an angle in a right-angled triangle is defined as:

$ \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} $

Substituting the values we found:

$ \sin(\theta) = \frac{5k}{13k} $

$ \sin(\theta) = \frac{5}{13} $

Thus, the value of the sine of the angle is $\frac{5}{13}$.