Problem 49 - Algebra 2

Question

Solve each equation. Round to the nearest tenth. $$ a^{2}=3^{2}+5^{2}-2(3)(5) \cos 85^{\circ} $$

Step-by-Step Solution

Answer

\( a \approx 5.6 \)

Step 1: Identify the Equation Type

The given equation is based on the Law of Cosines: \( c^2 = a^2 + b^2 - 2ab \cos(C) \). Here, \( c \) is represented by \( a \), \( a \) and \( b \) are the other sides of a triangle (3 and 5 in this case), and the angle \( C \) is 85°. So, the equation is already set to use the Law of Cosines.

Step 2: Plug Values into the Formula

Substitute the values into the Law of Cosines: \[ a^2 = 3^2 + 5^2 - 2(3)(5) \cos(85^{\circ}) \]evaluate it step-by-step.

Step 3: Calculate Each Component

First, calculate the squares of the sides:- \( 3^2 = 9 \)- \( 5^2 = 25 \)Next, evaluate the cosine term:- \( \cos(85^{\circ}) \approx 0.0872 \) (Using a calculator)

Step 4: Simplify the Equation

Now substitute the numbers back into the equation:\[ a^2 = 9 + 25 - 2(3)(5)(0.0872) \]Calculate\[- 2(3)(5)(0.0872) = 2.616 \]So,\[ a^2 = 9 + 25 - 2.616 \]

Step 5: Solve for a²

Simplify further:\[ a^2 = 34 - 2.616 = 31.384 \]

Step 6: Find a by Taking the Square Root

Calculate \( a \) by taking the square root:\[ a = \sqrt{31.384} \approx 5.6 \]This value is the length of side \( a \) rounded to the nearest tenth.

Key Concepts

Trigonometry

Trigonometry is the branch of mathematics that deals with the study of triangles, particularly right triangles. It explores the relationships between the angles and sides of triangles, using concepts such as sine, cosine, and tangent. These are functions that relate the angles of a triangle to the lengths of its sides.

In this exercise, we focused on the cosine function, which is an essential part of trigonometry. Understanding trigonometry helps us solve complex problems involving non-right triangles, which are triangles that don't necessarily have a 90-degree angle.

Triangle

A triangle is a three-sided polygon. It is one of the simplest shapes in geometry. Triangles can be classified by their side lengths and angles.

• Equilateral Triangle: All sides and angles are equal.
• Isosceles Triangle: Two sides and two angles are equal.
• Scalene Triangle: All sides and angles are different.

In addition to this classification based on sides, triangles can also be classified by their angles, such as acute, right, or obtuse triangles.

The Law of Cosines, which we used in this exercise, applies to any triangle. It allows us to calculate one side's length when we know two sides and the included angle.

Angle Measurement

An angle is formed by two rays meeting at a common endpoint called a vertex. In triangles, angles are crucial because they determine the shape and type of the triangle.

Angles in triangles are usually measured in degrees. There are 360 degrees in a complete circle, and the sum of angles in a triangle always equals 180 degrees.

In the given exercise, angle measurement comes into play when using the cosine function of 85°, which is crucial for applying the Law of Cosines. It's important to note that calculators should be set to degree mode to calculate trigonometric functions when angles are measured in degrees.

Cosine Function

The cosine function is one of the fundamental trigonometric functions relating the angle of a right triangle to the ratio of the adjacent side to the hypotenuse.

The cosine function is not limited to right triangles alone. It can also be used in non-right triangles to find unknown side lengths or angles, as demonstrated by the Law of Cosines. This law is an extended application of the cosine function in oblique triangles to relate their sides and angles.

In this problem, we found that the cosine of 85° is approximately 0.0872. This specific value of the cosine function plays a key role in calculating side lengths. Mastery over these calculations builds a solid foundation in trigonometry.