Problem 20 - Precalculus with Calculus Previews

Question

Use the Law of Sines to solve the triangle. $$ \alpha=75^{\circ}, \gamma=45^{\circ}, b=8 $$

Step-by-Step Solution

Answer

\(a\approx8.92\), \(c\approx6.54\), \(\beta=60^{\circ}\).

Step 1: Recognize Known Elements

We are given two angles, \( \alpha = 75^{\circ} \) and \( \gamma = 45^{\circ} \), and one side, \( b = 8 \). We need to find the third angle \( \beta \), and the sides \( a \) and \( c \).

Step 2: Calculate the Third Angle

Use the property that the sum of the angles in a triangle is \( 180^{\circ} \). Calculate \( \beta \) as follows: \[ \beta = 180^{\circ} - \alpha - \gamma = 180^{\circ} - 75^{\circ} - 45^{\circ} = 60^{\circ}. \]

Step 3: Apply the Law of Sines

The Law of Sines states: \[ \frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} = \frac{c}{\sin(\gamma)}. \] Now plug in the known values to measure the unknown sides.

Step 4: Solve for Side 'a'

Using the \[ \frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} \] relationship, plug in the known values: \[ \frac{a}{\sin(75^{\circ})} = \frac{8}{\sin(60^{\circ})}. \] This translates to \[ a = \frac{8 \sin(75^{\circ})}{\sin(60^{\circ})}. \] Calculate this value using a calculator: \[ a \approx \frac{8 \times 0.9659}{0.8660} \approx 8.92. \]

Step 5: Solve for Side 'c'

Using the \[ \frac{c}{\sin(\gamma)} = \frac{b}{\sin(\beta)} \] relationship, plug in the known values: \[ \frac{c}{\sin(45^{\circ})} = \frac{8}{\sin(60^{\circ})}. \] Solve for \( c \): \[ c = \frac{8 \sin(45^{\circ})}{\sin(60^{\circ})}. \] Calculate this value using a calculator: \[ c \approx \frac{8 \times 0.7071}{0.8660} \approx 6.54. \]

Key Concepts

Trigonometry

Trigonometry is the branch of mathematics that explores the relationships between the sides and angles of triangles. This field of study is crucial because it provides the tools needed to solve complex problems involving right and non-right triangles. Trigonometry uses functions such as sine, cosine, and tangent to relate angles to side lengths. These functions can help calculate unknown side lengths or angles, provide insights into angles of elevation or depression, and are foundational in fields ranging from engineering to physics. Understanding trigonometry empowers students to translate geometric relationships into algebraic equations, forming a bridge between geometry and algebra.

Solving Triangles

Solving triangles is about finding the unknown sides and angles when certain elements of a triangle are known. This often involves using one or more of the trigonometric laws, such as the Law of Sines or the Law of Cosines.

• The Law of Sines is particularly handy when you have two angles and one side (AAS or ASA conditions) or two sides and a non-included angle (SSA condition).
• The Law of Cosines is used for cases where two sides and the included angle are known (SAS) or when all three sides are known (SSS).

To solve a triangle using the Law of Sines, it is crucial first to calculate any unknown angles using the fact that the sum of all angles in a triangle is 180°. After determining all angles, the Law of Sines is deployed, allowing the calculation of missing side lengths. This method combines given elements with the properties of sine to reveal unknown measures, ensuring that all triangle sides and angles are known.

Triangle Angles

Triangle angles are essential components for solving triangles, and they follow the rule that their sum must be 180°. This rule helps in finding an unknown angle when the other two angles are provided. For example, if two angles are 75° and 45°, the third angle would be 60°, as you simply subtract the sum of the known angles from 180°: 180° - 75° - 45° = 60°. When working with triangle angle measures:

• Remember that acute angles are less than 90°.
• A right angle is exactly 90°, which forms the basis for right triangles.
• Obtuse angles are those greater than 90°, perhaps seen in obtuse triangles.

These angle distinctions are necessary when utilizing trigonometric laws, as they provide the foundational constraints and conditions necessary for calculations, ensuring angles are handled appropriately with respect to their trigonometric functions.