Sine Ratio in a Right Triangle: sin ∠DAC × DC/AC

Grades 9-10

Question: i want to see what happen when the sin(<dac) from the right triangle ABC is multiplied by dc/ac (ac is the hypotenuses)

To understand what happens when the sine of

\angle DAC

is multiplied by the ratio

\frac{DC}{AC}

, we can look at the geometry of the triangle through the Law of Sines and Area Formulas.

Consider a right triangle

\triangle ABC

where the right angle is at

B

(making

AC

the hypotenuse). If we place a point

D

on the segment

BC

, we divide the original triangle into two smaller triangles:

\triangle ABD

and

\triangle ADC

The Sine Rule in $\triangle ADC$ : In any triangle, the ratio of a side to the sine of its opposite angle is constant. For $\triangle ADC$ , this gives us:

$\frac{DC}{\sin(\angle DAC)} = \frac{AC}{\sin(\angle ADC)}$

Rearranging this formula to find $\sin(\angle DAC)$ :

$\sin(\angle DAC) = \frac{DC}{AC} \sin(\angle ADC)$
Evaluating the Expression: The problem asks for the value of $\sin(\angle DAC) \times \frac{DC}{AC}$ . Substituting the expression from step 1:

$\text{Result} = \left( \frac{DC}{AC} \sin(\angle ADC) \right) \times \frac{DC}{AC} = \left( \frac{DC}{AC} \right)^2 \sin(\angle ADC)$
Geometric Interpretation:
- Since $\angle ADC$ is an exterior angle to the right triangle $\triangle ABD$ , $\sin(\angle ADC) = \sin(\angle ADB) = \frac{AB}{AD}$ .
- Substituting this in, the expression becomes: $\frac{AB \cdot DC^2}{AD \cdot AC^2}$ .
- Special Case ( $D = B$ ): If the point $D$ moves all the way to vertex $B$ , then $DC$ becomes $BC$ , and $\angle DAC$ becomes the full angle $\angle BAC$ . In this case, the expression simplifies to $\sin(\angle BAC) \cdot \frac{BC}{AC}$ . Since $\sin(\angle BAC) = \frac{BC}{AC}$ , the result is simply $\sin^2(\angle BAC)$ .
- Special Case ( $D = C$ ): If $D$ moves to $C$ , the length $DC$ becomes $0$ , so the entire product becomes $0$ .

In the interactive diagram:

Triangle $ABC$ is shown in gray, with the right angle at $B$ and the hypotenuse $AC$ in orange.
Point $D$ (blue) can be moved along the side $BC$ using the slider.
The Angle $\angle DAC$ ( $\alpha$ ) is marked at vertex $A$ .
The live calculation shows how the product changes as $D$ moves. You will notice that the value is always between $0$ and $\sin^2(\angle BAC)$ .