Question

In blood spatter analysis, the width, \(w,\) and the length, \(L,\) of a blood stain from a blood droplet are related to the impact angle, \(\theta,\) by the equation \(\sin \theta=\frac{w}{L}\) If \(\tan \theta=\frac{17}{10}\) and the width of a droplet stain is \(4 \mathrm{~mm},\) what is the approximate length of the stain?

Short answer

Approximate length: \(\text }}}'' or ~4.7}}}

Step-by-step solution

Understanding the relationship between trigonometric functions

First, recall that \(\tan \theta = \frac{\text{opposite}}{\text{adjacent}}\) and \(\tan \theta = \frac{17}{10}\) means the opposite side is 17 units, while the adjacent side is 10 units.

Finding \(\theta\) using \(\tan \theta\)

Calculate \(\theta\) by taking the inverse tangent: \(\theta = \tan^{-1} \frac{17}{10}\). Using a calculator, \(\theta \approx 59.74^{\text{o}}\).

Convert \(\theta\) into radians if needed

To proceed, use the angle in degrees unless a radians conversion is necessary.

Relation between width, length, and impact angle

Given \(\theta\) we also know that \(\theta\) is already known. Substitute in the formula: \(\theta = 59.74^{\text{o}}\).

Using the sine function

Remember that \(\theta\) remains in the degrees mode. The problem provides \(\theta\) in tangent, and the width and the length are calculated using Adding a tremendous sine equation. Once defined, let's calculate the sine degree.

Solving for L

Thus solving the equation directly, \(\theta=\frac{4}{L}\) transformed with the substitution: \(\text{sin} = \frac{w}{L}\) using \(\text{ above results}\rightarrow L = \frac{4}{\text{sin 59.74}\)

Key concepts

Sine Function

In trigonometry, the sine function, often written as \(\text{sin} \theta\), is one of the primary trigonometric functions. It relates the angle \( \theta \) to the ratio of the length of the opposite side of a right triangle to the hypotenuse. The function is defined as: \[ \text{sin} \theta = \frac{\text{opposite}}{\text{hypotenuse}} \]. For example, if we have a right triangle where the opposite side is 17 units and the hypotenuse is L, and we know \( \text{sin} \theta = \frac{w}{L} = \frac{4}{L}\), we use the angle \( \theta \) to find the length L. By knowing \( \theta = 59.74^\text{o} \), calculated from the tangent function, we can then solve: \[ L = \frac{4}{\text{sin } 59.74^\text{o}} \].

Tangent Function

The tangent function, \(\text{tan} \theta\), relates the angle \( \theta \) to the ratio of the length of the opposite side to the adjacent side in a right triangle. It is defined as: \[ \text{tan} \theta = \frac{\text{opposite}}{\text{adjacent}} \]. In the given exercise, we know \( \text{tan} \theta = \frac{17}{10} \), which tells us the opposite side is 17 units long and the adjacent side is 10 units. To find \( \theta \), we take the inverse of the tangent function: \[ \theta = \tan^{-1} \frac{17}{10} \]. Plugging this into a calculator, we get \( \theta \) to be approximately \( 59.74^\text{o} \). This value of \( \theta \) is then used in the sine equation to find the length of the bloodstain.

Inverse Trigonometric Functions

Inverse trigonometric functions are used to determine the angles when given the ratio of sides. For example, if we know the value of \( \text{tan} \theta \), we use the inverse tangent function, written as \( \tan^{-1}\), to find \( \theta \). In the exercise, we used \( \theta = \tan^{-1} \frac{17}{10} \), which gave us \( \theta \) as approximately \( 59.74^\text{o} \). Similarly, we can use \( \text{sin}^{-1}, \text{cos}^{-1},\) and other inverse trigonometric functions to find individual angles when other side ratios are known. Understanding how to switch between the functions and their inverses is crucial in applying them to real-world problems such as blood spatter analysis.

Blood Spatter Analysis

Blood spatter analysis is a forensic method used to reconstruct the events of a crime involving bloodshed. It involves examining the shapes, sizes, and distribution of bloodstains to determine the angle and velocity at which blood droplets hit a surface. In the provided exercise, we analyzed the relationship between the width and length of a bloodstain using the impact angle. The sine function \[ \text{sin} \theta = \frac{w}{L} \] helped relate the angle \( \theta \) to the dimensions of the stain. Given \( \theta \) from the tangent function and the width, we solved for the length to understand the droplet's path and impact. This analysis aids forensic investigators in piecing together the sequence of events in a crime scene.