Problem 1 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-8: Given the indicated parts of triangle \(A B C\) with \(\gamma=90^{\circ}\), find the exact values of the remaining parts. $$ \alpha=30^{\circ}, \quad b=20 $$

Step-by-Step Solution

Answer

\(\beta = 60^{\circ}\), \(c = 40\), \(a = 20\sqrt{3}\).

Step 1: Use the Triangle Angle Sum Property

In any triangle, the sum of the angles is always \(180^{\circ}\). Given \(\gamma = 90^{\circ}\) and \(\alpha = 30^{\circ}\), we can find \(\beta\) using the formula:\[ \beta = 180^{\circ} - \gamma - \alpha = 180^{\circ} - 90^{\circ} - 30^{\circ} = 60^{\circ} \]

Step 2: Use the Sine Function to Find Side 'a'

In a right triangle, the sine of an angle is the ratio of the opposite side to the hypotenuse. With \(\alpha = 30^{\circ}\) and side 'b' as the opposite side:\[ \sin(\alpha) = \frac{b}{c} \quad \Rightarrow \quad \sin(30^{\circ}) = \frac{20}{c} \]\(\sin(30^{\circ}) = \frac{1}{2}\), so:\[ \frac{1}{2} = \frac{20}{c} \quad \Rightarrow \quad c = 40 \]

Step 3: Use the Cosine Function to Find Side 'a'

With side 'c' as the hypotenuse and \(\alpha = 30^{\circ}\):\[ \cos(\alpha) = \frac{a}{c} \quad \Rightarrow \quad \cos(30^{\circ}) = \frac{a}{40} \]\(\cos(30^{\circ}) = \frac{\sqrt{3}}{2}\), so:\[ \frac{\sqrt{3}}{2} = \frac{a}{40} \quad \Rightarrow \quad a = 40 \times \frac{\sqrt{3}}{2} = 20\sqrt{3} \]

Key Concepts

Right Triangle

A right triangle is a special type of triangle where one of the angles is exactly 90 degrees, known as a right angle. The right angle is typically denoted by the Greek letter \(\gamma\). This angle divides the triangle into two parts: the two legs and the hypotenuse.
The hypotenuse is the longest side of the right triangle, always opposite the right angle. Meanwhile, the other two sides are called the legs, which can be of varying lengths.
A common characteristic of right triangles is the Pythagorean theorem, but in this exercise, we focus more on the interplay of angles and side lengths through trigonometric functions.

Sine Function

The sine function is one of the fundamental trigonometric functions that relates the angle in a right triangle to the ratio of the lengths of specific sides of the triangle. To be specific, the sine of an angle \(\alpha\) (not the right angle) is defined as:
\[ \sin(\alpha) = \frac{\text{opposite side}}{\text{hypotenuse}} \]
In our exercise, \(\alpha = 30^\circ \), and the side opposite this angle (\(b\)) is 20. The sine of 30 degrees is 0.5, or \(\frac{1}{2}\). Thus, using the equation:

• \(\sin(30^\circ) = \frac{b}{c} \)
• \(\frac{1}{2} = \frac{20}{c}\)

We can solve for \(c\), the hypotenuse, giving us \(c = 40\).

Cosine Function

The cosine function is another key trigonometric function that involves the ratio of the adjacent side to the hypotenuse in a right triangle. For an angle \(\alpha\), the cosine is given by:
\[ \cos(\alpha) = \frac{\text{adjacent side}}{\text{hypotenuse}} \]
In this scenario, since \(\alpha = 30^\circ\), we know that the cosine values are consistent, and the adjacent side \(a\) is what we want to determine. We use the equation:

• \(\cos(30^\circ) = \frac{a}{40}\)

The cosine of 30 degrees is \(\frac{\sqrt{3}}{2}\). Solving this produces:
\(a = 40 \times \frac{\sqrt{3}}{2} = 20\sqrt{3}\).

Angle Sum Property

The angle sum property of triangles states that the sum of the angles in any triangle is always \(180^\circ\). This rule is vital for finding the third angle in a right triangle when the other two are known. In a right triangle,
\(\gamma = 90^\circ\), and in our exercise, \(\alpha = 30^\circ\). We find the remaining angle by:

• \(\beta = 180^\circ - \gamma - \alpha\)
• \(\beta = 180^\circ - 90^\circ - 30^\circ = 60^\circ\)

This confirms that the angles fit the required property, ensuring the triangle is valid.

Trigonometric Ratios

Trigonometric ratios are simple yet powerful mathematical tools that relate the angles of a right triangle to its side lengths. There are three fundamental trigonometric functions: sine, cosine, and tangent.
In this exercise, the focus is on the sine and cosine functions, each describing a specific relationship:

• Sine function: relates the opposite side to the hypotenuse.
• Cosine function: relates the adjacent side to the hypotenuse.

By using these ratios, we can solve for unknown side lengths or angles when given partial information. They are crucial in solving right triangles and many problems in trigonometry.