Problem 31 - Calculus for Biology and Medicine

Question

A triangle has vertices at coordinates \(P=(0,0), Q=(4,0)\), and \(R=(4,3)\). (a) Use basic trigonometry to compute the lengths of all three sides and the measures of all three angles. (b) Use the results of this section to repeat (a).

Step-by-Step Solution

Answer

Lengths: PQ = 4, QR = 3, PR = 5; Angles: P ≈ 36.87°, Q ≈ 26.57°, R ≈ 116.56°.

Step 1: Calculate Side Lengths Using Distance Formula

To find the length of side PQ, use the distance formula: \[ \text{Length of } PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]Substitute the coordinates of P and Q:\[ \text{Length of } PQ = \sqrt{(4 - 0)^2 + (0 - 0)^2} = \sqrt{16} = 4 \]Similarly, calculate QR:\[ \text{Length of } QR = \sqrt{(4 - 4)^2 + (3 - 0)^2} = \sqrt{9} = 3 \]For PR:\[ \text{Length of } PR = \sqrt{(4 - 0)^2 + (3 - 0)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

Step 2: Use Cosine Rule to Find Angles

To find the angle at vertex P (i.e., \( \angle PQR \)), apply the cosine rule:\[ \cos(\angle P) = \frac{b^2 + c^2 - a^2}{2bc} \]Where \( a = 3 \), \( b = 5 \), and \( c = 4 \):\[ \cos(\angle P) = \frac{4^2 + 5^2 - 3^2}{2 \times 4 \times 5} = \frac{16 + 25 - 9}{40} = \frac{32}{40} = 0.8 \]Thus, \( \angle P = \cos^{-1}(0.8) \approx 36.87^\circ \).

Step 3: Calculate Second Angle Using Sine Rule

Use the sine rule for \( \angle Q \):\[ \frac{\sin(\angle Q)}{a} = \frac{\sin(\angle P)}{c} \]\[ \frac{\sin(\angle Q)}{3} = \frac{\sin(36.87^\circ)}{4} \]\[ \sin(\angle Q) = 3 \times \frac{0.6}{4} = 0.45 \]\( \angle Q = \sin^{-1}(0.45) \approx 26.57^\circ \).

Step 4: Determine Third Angle by Subtraction

Knowing that the sum of angles in a triangle is 180 degrees, find \( \angle R \):\[ \angle R = 180^\circ - \angle P - \angle Q \]\[ \angle R = 180^\circ - 36.87^\circ - 26.57^\circ \approx 116.56^\circ \].

Key Concepts

Understanding the Distance Formula

To start, the distance formula helps us find the length between two points on a coordinate plane. This formula comes in particularly handy in biology when calculating distances between points, such as in genetic sequences or cellular arrays. The formula is written as:

• \( \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)

For the triangle with vertices at \(P=(0,0)\), \(Q=(4,0)\), and \(R=(4,3)\), applying the distance formula allows us to measure each of the sides accurately. This involves substituting the coordinates of each point pair into the formula. When we plug in the points for side \(PQ\), for example, the calculation simplifies to \(\sqrt{(4-0)^2 + (0-0)^2}\), culminating in a length of 4. Repeating this for \(QR\) and \(PR\) reveals their lengths as 3 and 5, respectively. Breaking the coordinates into their respective \(x\) and \(y\) components is essential for precise calculations in biology-related fields, as it aids in visual accuracy.

Applying the Cosine Rule

The cosine rule is used to find unknown angles or sides in any triangle, especially when traditional right-angle trigonometry can't be applied. The rule is written as:

• \( c^2 = a^2 + b^2 - 2ab \cos(C) \)

For our problem, we use it to determine the angle \(\angle P\) within triangle \(PQR\). After determining side lengths, the cosine rule turns the relation between sides and angles into a calculable form. With \(PQ = 4\), \(QR = 3\), and \(PR = 5\), substituting into \( \cos(C) = \frac{c^2 + a^2 - b^2}{2ac} \) provides a clean approach to deducing \(\cos^{-1}(0.8)\). The result, around \(36.87^\circ\), represents a crucial aspect of biometry studies where angles need determining, facilitating tasks like calculating angles between gene regulatory networks.

Exploring the Sine Rule

The sine rule allows us to find unknown angles and sides in any triangle when we have sufficient information. It's expressed as:

• \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \)

To deduce \(\angle Q\), relating it to a known angle like \(\angle P\) is useful. From earlier calculations, we found \(\angle P\) was approximately \(36.87^\circ\). To find \(\angle Q\), fill into the equation \(\frac{\sin(\angle Q)}{3} = \frac{\sin (36.87^\circ)}{4}\). Simplifying further, we find \(\sin(\angle Q) = 0.45\), providing a means to obtain \(\angle Q \approx 26.57^\circ\). This technique aids in discovering angles in diverse biological fields where molecular structures are examined.

Understanding Angles in a Triangle

Angles in a triangle always equate to 180 degrees, a fundamental property used to compute unknown angles. For triangle \(PQR\) with known angles \(P\) and \(Q\), subtract their total from 180 to find \(\angle R\). This sum consistency simplifies computations without measuring every angle directly. In our scenario:

• \(\angle R = 180^\circ - 36.87^\circ - 26.57^\circ\)
• \(\angle R \approx 116.56^\circ\)

The property plays pivotal roles in sectors like bioinformatics and phylogenetics, where organisms sometimes relate under triangular frameworks. Recognizing equalities tightly holds problem-solving composure and streamlining.Harnessing these concepts align science and math, highlighted by triangle calculations.

The Role of Coordinate Geometry

Coordinate geometry precisely describes shapes and figures using algebraic expressions. It's crucial when pinpointing positions in plane figures seen in biology, like cellular models. Triangle \(PQR\)—positioned by \((0,0)\), \((4,0)\), and \((4,3)\)—exemplifies this.Using coordinates not only locates points but applies across disciplines. Consider an octopus' tentacles, trackable via coordinates for movements observed in marine research. The interplay of calculus, algebra, and geometry assists data organization, supporting insightful discovery and biological understanding. Such applications ground research with coordinate precision, harmonizing mathematics with organic phenomena.