Problem 9 - Precalculus: Functions and Graphs

Question

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

\( \beta = 53^{\circ}, \ c \approx 14.44, \ a \approx 19.17 \)

Step 1: Identify the Given Components

We are given \( \triangle ABC \) where \( \angle C = 90^{\circ} \), \( \alpha = \angle A = 37^{\circ} \), and \( b = AC = 24 \). Our goal is to find the remaining parts, namely \( \beta = \angle B \), \( a = AB \), and \( c = BC \).

Step 2: Calculate \( \beta \)

Since the angles in a triangle sum up to \(180^{\circ}\), and one of the angles is \(90^{\circ}\), we can find \(\beta\) by subtracting the known angles from \(90^{\circ}\): \[ \beta = 90^{\circ} - \alpha = 90^{\circ} - 37^{\circ} = 53^{\circ}. \]

Step 3: Calculate the Length of \( c \)

We can use the sine function to find \(c\), the side opposite \(\alpha\): \[ \sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{c}{b} \] Substitute the known values: \[ \sin(37^{\circ}) = \frac{c}{24} \] \[ c = 24 \cdot \sin(37^{\circ}) \] Using \( \sin(37^{\circ}) \approx 0.6018\), we find: \[ c \approx 24 \cdot 0.6018 = 14.4432. \] So \( c \approx 14.44 \).

Step 4: Calculate the Length of \( a \)

Next, use the cosine function to find \(a\), the side adjacent to \(\alpha\): \[ \cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{a}{b} \] Substituting the known values: \[ \cos(37^{\circ}) = \frac{a}{24} \] \[ a = 24 \cdot \cos(37^{\circ}) \] Using \( \cos(37^{\circ}) \approx 0.7986\), we find: \[ a \approx 24 \cdot 0.7986 = 19.1664. \] So \( a \approx 19.17 \).

Key Concepts

Triangle Angle Sum

In any triangle, all three interior angles add up to a total of 180 degrees. This is known as the Triangle Angle Sum Property. For a right triangle, one of these angles is always 90 degrees. Thus, the sum of the other two angles must be 90 degrees.For triangle \( \triangle ABC \) with angle \( \gamma = 90^{\circ} \), and \( \alpha = 37^{\circ} \), we want to find angle \( \beta \). Simply subtract \( \alpha \) from 90 degrees because \( \beta = 90^{\circ} - \alpha \). Consequently, angle \( \beta \) equals 53 degrees, completing the triangle's angle measures and helping us solve for the unknown sides.

Sine Function

The sine function is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of the length of the side opposite the angle over the length of the hypotenuse.When calculating the side opposite \( \alpha \) in \( \triangle ABC \), we use the sine function:- Formula: \( \sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}} \).Given \( \alpha = 37^{\circ} \) and \( b = 24 \) (the hypotenuse),- Substitute the values: \( \sin(37^{\circ}) = \frac{c}{24} \).- Solving, we get: \( c = 24 \cdot \sin(37^{\circ}) \). Using the approximate value \( \sin(37^{\circ}) \approx 0.6018 \), side \( c \approx 14.44 \). This process leverages the power of the sine function to find unknown side lengths in right triangles.

Cosine Function

The cosine function relates a particular angle in a right triangle to the ratio of the length of the side adjacent to the angle over the length of the hypotenuse.In \( \triangle ABC \), where we need to find the length of the side adjacent to \( \alpha = 37^{\circ} \), which is \( a \), we use:- Formula: \( \cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}} \).Since \( b = 24 \) is the hypotenuse,- Substitute to get: \( \cos(37^{\circ}) = \frac{a}{24} \).- Solving, \( a = 24 \cdot \cos(37^{\circ}) \). With \( \cos(37^{\circ}) \approx 0.7986 \), the calculated side length \( a \approx 19.17 \). This exemplifies how the cosine function helps find unknown sides when the angle and the hypotenuse are known.

Trigonometric Ratios

Trigonometric ratios are invaluable tools in solving right triangles. These include the sine, cosine, and tangent functions, which compare different side lengths of a triangle.- **Sine ratio** compares the opposite side to the hypotenuse.- **Cosine ratio** compares the adjacent side to the hypotenuse.- **Tangent ratio** compares the opposite side to the adjacent side.In our example, we used the sine and cosine functions to solve \( \triangle ABC \) by finding the lengths of sides \( a \) and \( c \). These trigonometric ratios allow us to compute angles and side measurements efficiently and are foundational concepts in trigonometry. Understanding these relationships simplifies problems involving right triangles, making them easier to solve.