Problem 63 - Algebra 2

Question

PREREQUISITE SKILL Solve each equation. Round to the nearest tenth. $$ \frac{a}{\sin 32^{\circ}}=\frac{8}{\sin 65^{\circ}} $$

Step-by-Step Solution

Answer

The value of \( a \) is approximately 4.7.

Step 1: Understand the Equation

The given equation is \( \frac{a}{\sin 32^{\circ}}=\frac{8}{\sin 65^{\circ}} \). This is a common form encountered in the law of sines, which relates the sides of a triangle to the sines of its angles.

Step 2: Isolate the Variable

To solve for \( a \), multiply both sides of the equation by \( \sin 32^{\circ} \). This gives us \( a = \frac{8}{\sin 65^{\circ}} \times \sin 32^{\circ} \).

Step 3: Substitute Known Values

Calculate the sine of 32 degrees and 65 degrees using a calculator: \( \sin 32^{\circ} \approx 0.5299 \) and \( \sin 65^{\circ} \approx 0.9063 \).

Step 4: Perform the Calculation

Substitute these values back into the equation: \( a = \frac{8}{0.9063} \times 0.5299 \). Calculate \( \frac{8}{0.9063} \approx 8.826 \) and then multiply by 0.5299 to get \( a \approx 4.68 \).

Step 5: Round to the Nearest Tenth

Round 4.68 to the nearest tenth. The value of \( a \) is approximately 4.7.

Key Concepts

Trigonometric Equations

Trigonometric equations involve relationships between angles and sides of triangles using trigonometric functions like sine, cosine, and tangent. These equations are essential for solving problems in trigonometry, often represented as equalities that incorporate these functions.
One common trigonometric equation arises from the Law of Sines. This equation relates the sides of a triangle to the sines of its opposite angles, as shown in the example: \( \frac{a}{\sin 32^{\circ}}=\frac{8}{\sin 65^{\circ}} \).
This type of equation requires understanding how to manipulate sines and angles to isolate and solve for unknowns, such as the length of a triangle's side. By isolating the variable \(a\) and using known sine values, the equation can be solved numerically. Key skills include:

• Identifying known values and substituting them into the equation.
• Manipulating the equation algebraically to isolate variables.
• Evaluating trigonometric functions using a calculator for accuracy.

Sine Function

The sine function is one of the primary trigonometric functions, pivotal in studying and solving triangles. It relates an angle in a right triangle to the ratio of the length of the opposite side to the hypotenuse.
This is crucial when applying the Law of Sines, as it allows one to find missing sides and angles by using the known values of other sides or angles. When dealing with an equation like \( \frac{a}{\sin 32^{\circ}}=\frac{8}{\sin 65^{\circ}} \), the sine values \( \sin 32^{\circ} \) and \( \sin 65^{\circ} \) play a fundamental role.
Using a scientific calculator, these sine values can be found accurately:

• \( \sin 32^{\circ} \approx 0.5299 \)
• \( \sin 65^{\circ} \approx 0.9063 \)

These calculations must be precise to ensure an accurate result, especially when working with more complex problems beyond simple triangles.

Triangle Geometry

Triangle geometry fundamentally explores the properties and relationships within triangles. Triangles can vary by angle sizes and side lengths, making problems unique based on their given parameters. Concepts like the Law of Sines come into play in non-right triangles, where we need to understand:

• The relationship between sides and their opposite angles.
• How to calculate unknown angles or sides using known measures.
• Differentiating between types of triangles (e.g., scalene, isosceles).

In practical application, such as in the provided problem involving \( \frac{a}{\sin 32^{\circ}}=\frac{8}{\sin 65^{\circ}} \), triangle geometry offers insight into solving and understanding the equation.
By visualizing the triangle, the roles of the specified angles and their opposite sides help apply trigonometric equations effectively. Knowing these geometrical relationships aids in accurately transforming abstract equations into tangible solutions.