Problem 4 - Algebra 2 and Trigonometry

Question

In \(\triangle P Q R,\) express cos \(Q\) in terms of \(p, q,\) and \(r\)

Step-by-Step Solution

Answer

\(\cos(Q) = \frac{p^2 + r^2 - q^2}{2pr}\)

Step 1: Understand the Problem

We need to express the cosine of angle \(Q\) in the triangle \(\triangle PQR\) in terms of its side lengths \(p, q,\) and \(r.\) Here, \(p, q,\) and \(r\) are the lengths of the sides opposite to angles \(P, Q,\) and \(R,\) respectively.

Step 2: Recall the Cosine Rule

The cosine rule in a triangle is given by:\[c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\where \(a, b,\) and \(c\) are the lengths of the sides, and \(C\) is the angle opposite side \(c.\)\]

Step 3: Apply the Cosine Rule to \(\triangle PQR\)

In \(\triangle PQR,\) set \(a = p, b = r,\) and \(c = q.\) Applying the cosine rule: \[q^2 = p^2 + r^2 - 2pr \cdot \cos(Q)\] which expresses \(\cos(Q)\) in terms of \(p, q,\) and \(r.\)

Step 4: Solve for \(\cos(Q)\)

Rearrange the equation to solve for \(\cos(Q):\)\[\cos(Q) = \frac{p^2 + r^2 - q^2}{2pr}\]

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics that focuses on the study of triangles. It greatly explores the relationships between angles and the lengths of sides in triangles. This field is essential for solving problems involving angles and dimensions in a triangle.

In trigonometry, the primary focus is on right-angled triangles, but it extends to non-right-angled triangles as well. The techniques developed in this area have applications in fields such as physics, engineering, astronomy, and even in art and architecture.

• The three primary trigonometric ratios are sine, cosine, and tangent. These relate an angle of a triangle to the ratios of two of its sides.
• Trigonometry helps in calculating angles in situations where direct measurement isn’t feasible, using known side lengths.

Triangle Sides

In every triangle, there are three sides. In the context of trigonometry, understanding the relationship between these sides is crucial.

This exercise involves a triangle, \(\triangle PQR\), with sides labeled as \(p, q,\) and \(r.\) Each of these sides is associated with a specific angle:

• Side \(p\) is opposite angle \(P.\)
• Side \(q\) is opposite angle \(Q.\)
• Side \(r\) is opposite angle \(R.\)

The understanding of these sides, and their lengths, helps in applying trigonometric rules, like the cosine rule, to find unknown angles or solve other geometric problems.

Cosine of Angle

The cosine of an angle in a triangle is one of the fundamental trigonometric functions, represented by \(\cos(\theta)\). This function takes an angle and gives you a ratio of the length of the adjacent side to the hypotenuse if it's a right-angled triangle. However, it can also be used in any triangle using the cosine rule.

In the exercise involving triangle \(\triangle PQR\), we use the cosine rule to find the cosine of angle \(Q\). The cosine rule redefines cosine functions for any triangle, allowing you to express cosine in terms of the sides of the triangle. Here is the specific rule used:\[\cos(Q) = \frac{p^2 + r^2 - q^2}{2pr}\]

This equation shows how the cosine of angle \(Q\) is derived using the lengths of the sides \(p, q,\) and \(r\). This formula is especially helpful in exploring non-right triangles where the standard trigonometric identities don't apply directly.

Angle in Triangle

Angles in a triangle are critical to understanding the triangle's properties. All triangles have three angles, and the sum of these angles is always 180 degrees.

In a triangle like \(\triangle PQR\), angles \(P, Q,\) and \(R\) are the internal angles we often solve for using trigonometric identities and formulas. Understanding these angles allows us to explore more complex geometric relationships and solve problems precisely:

• Each angle is opposite to a specific side, which is essential for applying trigonometric rules effectively.
• The relationship of angles and sides in a triangle is expressed through formulas like the sine and cosine rules, which can compute unknown sides or angles.

Using the cosine rule, as shown in the exercise, allows for expressing an angle in terms of the triangle's side lengths, giving a more profound understanding of its geometry.