Problem 62 - Precalculus

Question

Suppose that you have a triangle with side lengths \(a, b,\) and \(c,\) and angles \(\alpha, \beta,\) and \(\gamma,\) respectively, directly across from them. If it is known that \(a=\frac{1}{\sqrt{2}} b, c=2, \alpha\) is an acute angle, and \(\beta=2 \alpha,\) solve the triangle.

Step-by-Step Solution

Answer

The triangle is a right triangle with sides \(a = 2\), \(b = 2\sqrt{2}\), \(c = 2\), and angles \(\alpha = 45^\circ\), \(\beta = 90^\circ\), \(\gamma = 45^\circ\).

Step 1: Analyzing Triangular Relationships

Given that \( a = \frac{1}{\sqrt{2}}b \), \( c = 2 \), and \( \beta = 2\alpha \), we recognize this situation involves known relations between angles and sides in a triangle. Our goal is to determine \( a, b, c, \alpha, \beta, \) and \( \gamma \).

Step 2: Using the Law of Sines

Apply the Law of Sines, \( \frac{a}{\sin \alpha} = \frac{b}{\sin \beta} = \frac{c}{\sin \gamma} \). With relations \( a = \frac{1}{\sqrt{2}}b \) and \( \beta = 2\alpha \), we have \( \frac{1}{\sqrt{2}}b = b \cdot \frac{\sin \alpha}{\sin 2\alpha} \). Since \( \sin 2\alpha = 2 \sin \alpha \cos \alpha \), simplify to find an expression for \( \cos \alpha \).

Step 3: Solving for \(\cos \alpha\)

With \( \frac{1}{\sqrt{2}} = \frac{\sin \alpha}{2 \sin \alpha \cos \alpha} \), simplify to \( \sqrt{2} = 2 \cos \alpha \). Therefore, \( \cos \alpha = \frac{1}{\sqrt{2}} \), which implies \( \alpha = 45^\circ \) as \( \alpha \) must be acute.

Step 4: Finding \(\beta\) and \(\gamma\)

Knowing \( \alpha = 45^\circ \), calculate \( \beta = 2\alpha = 90^\circ \). Since the angle sum of a triangle is \( 180^\circ \), \( \gamma = 180^\circ - 90^\circ - 45^\circ = 45^\circ \).

Step 5: Finalizing Side Lengths

Since \( \alpha = \gamma = 45^\circ \), by symmetry and using the Law of Sines, \( a = c \). Thus, \( a = 2 \) and \( b = 2\sqrt{2} \) (from the relation \( a = \frac{1}{\sqrt{2}}b \) with \( a = 2 \)).

Key Concepts

Law of Sines

The Law of Sines is a powerful tool in trigonometry for solving triangles. It relates the lengths of the sides of a triangle to the sines of its angles. This law is written as:

• \( \frac{a}{\sin \alpha} = \frac{b}{\sin \beta} = \frac{c}{\sin \gamma} \)

In any triangle, whether acute, right, or obtuse, the Law of Sines allows us to find unknown angles or sides if we have enough information.

In the given exercise, we use this law to explore the relationships between the sides and angles. The problem involves a specific condition where one angle is twice the other (\( \beta = 2\alpha \)). This gives us an opportunity to apply the Law of Sines to develop an equation that will help us find the unknown angles \( \alpha \), \( \beta \), and \( \gamma \).

By substituting and simplifying with known identities, such as \( \sin 2\alpha = 2 \sin \alpha \cos \alpha \), we can solve for particular values, such as \( \cos \alpha \). This calculated value helps us deduce other angles, showcasing how the Law of Sines is an essential method for triangulation.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variables involved. They are essential tools for solving problems in precalculus and are frequently used to simplify expressions or solve equations.

For instance, in the exercise, one important trigonometric identity used is the double angle formula:

• \( \sin 2\alpha = 2 \sin \alpha \cos \alpha \)

This identity is crucial when the angle relationship \( \beta = 2\alpha \) exists, as shown in the problem. It helps simplify the expression and find the value of \( \cos \alpha \).

Trigonometric identities like the double angle formulas and Pythagorean Identities \( \sin^2\theta + \cos^2\theta = 1 \) are frequently used together, creating systems or simplifying expressions to unravel more complex trigonometric problems. Knowing these can provide a significant advantage in understanding and solving various triangle and angle-related questions.

Angle Sum Property

The Angle Sum Property is a fundamental concept in geometry, particularly for triangles. It states that the sum of the angles in any triangle is always \( 180^\circ \).

This principle plays a vital role in solving the given triangle. Once we determine \( \alpha \) and \( \beta \), we can find \( \gamma \) by using this property. As per the Angle Sum Property, if \( \alpha = 45^\circ \) and \( \beta = 90^\circ \), then \( \gamma = 180^\circ - (45^\circ + 90^\circ) = 45^\circ \).

This property is not only essential for calculating missing angles but also for verifying the correctness of computed values. Understanding the Angle Sum Property ensures that all calculated angles in a triangle adhere to the rule of adding up to \( 180^\circ \), providing a consistent check when solving problems in trigonometry and geometry.