Problem 9 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 9-16: Given the indicated parts of triangle \(A B C\) with \(\gamma=90^{\circ}\), approximate the remaining parts. $$ \alpha=37^{\circ}, \quad b=24 $$

Step-by-Step Solution

Answer

\( \beta = 53^\circ \), \( a \approx 18.08 \), \( c \approx 30.03 \).

Step 1: Identify Known Parts

We are given that triangle \( ABC \) is a right triangle with the right angle \( \gamma = 90^\circ \). We also know the angle \( \alpha = 37^\circ \) and the length \( b = 24 \).

Step 2: Calculate Angle \( \beta \)

Since the sum of angles in a triangle is \( 180^\circ \), we can find \( \beta \) using the equation:\[\beta = 180^\circ - \gamma - \alpha = 180^\circ - 90^\circ - 37^\circ = 53^\circ.\]

Step 3: Use Trigonometric Ratios to Find \( a \)

In a right triangle, \( \tan \alpha = \frac{\text{opposite}}{\text{adjacent}} = \frac{a}{b} \). We can find \( a \) using:\[a = b \cdot \tan \alpha = 24 \cdot \tan 37^\circ.\]Use a calculator to find \( \tan 37^\circ \approx 0.7535 \), so:\[a = 24 \times 0.7535 \approx 18.08.\]

Step 4: Calculate Side \( c \) Using Pythagorean Theorem

Using the Pythagorean theorem \( c^2 = a^2 + b^2 \), we can find \( c \):\[c^2 = 18.08^2 + 24^2 \approx 326.24 + 576 = 902.24,\]\[ c = \sqrt{902.24} \approx 30.03.\]

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry, particularly in the context of right triangles. It states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.To apply this theorem, let's consider a right triangle where the sides are labeled as follows:

• Side \(a\)
• Side \(b\)
• Hypotenuse \(c\)

According to the Pythagorean Theorem, the equation is:\[c^2 = a^2 + b^2\]In our exercise, we found the side lengths by using trigonometric ratios, then applied this theorem to find the hypotenuse \(c\). By knowing \(a = 18.08\) and \(b = 24\), we calculated \(c\) as follows:\[c^2 = 18.08^2 + 24^2 \approx 902.24\]Hence, \(c \approx \sqrt{902.24} \approx 30.03\). The theorem provides a robust tool for connecting the sides of a right triangle through simple arithmetic.

Trigonometric Ratios

Trigonometric Ratios are key to understanding and solving problems in right triangle trigonometry. These ratios relate the angles of a triangle to the lengths of its sides.The primary trigonometric ratios include:

• Sine: \(\sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}}\)
• Cosine: \(\cos \theta = \frac{\text{adjacent side}}{\text{hypotenuse}}\)
• Tangent: \(\tan \theta = \frac{\text{opposite side}}{\text{adjacent side}}\)

In our exercise, we utilized the tangent ratio because we were given an angle \(\alpha = 37^\circ\) and needed to find the side length \(a\).The formula used was:\[a = b \cdot \tan \alpha = 24 \cdot \tan 37^\circ\]By calculating \(\tan 37^\circ\) as approximately 0.7535, we determined \( a \approx 18.08 \). Understanding these ratios allows us to link angle measures to side lengths, a crucial skill in trigonometry.

Angle Sum Property

The Angle Sum Property of a triangle dictates that the sum of a triangle's interior angles always equals \(180^\circ\). This rule is especially useful in right triangle problems.For a right triangle:

• One angle is always \(90^\circ\)
• The other two angles must add up to \(90^\circ\)

In the given exercise, knowing \(\gamma = 90^\circ\) and \(\alpha = 37^\circ\) allowed us to easily calculate the third angle, \(\beta\). This was done using the equation:\[\beta = 180^\circ - \gamma - \alpha = 180^\circ - 90^\circ - 37^\circ = 53^\circ\]This property helps to solve for unknown angles and is an essential part of triangle geometry, providing a straightforward way to find any missing interior angle once the others are known.