Problem 50 - Algebra 2

Question

Solve each equation. Round to the nearest tenth. $$ c^{2}=12^{2}+10^{2}-2(12)(10) \cos 40^{\circ} $$

Step-by-Step Solution

Answer

The value of \( c \) is approximately 7.8.

Step 1: Recognize the Law of Cosines

The equation given is a form of the Law of Cosines, which is used to find a side in a triangle when two sides and the included angle are known. The formula is: \( c^2 = a^2 + b^2 - 2ab \cos C \). In this problem, \( a = 12 \), \( b = 10 \), and \( C = 40^{\circ} \).

Step 2: Plug Values into the Equation

Insert the known values into the Law of Cosines equation: \[ c^2 = 12^2 + 10^2 - 2 \times 12 \times 10 \times \cos 40^{\circ} \].

Step 3: Calculate Squares of Known Sides

Calculate \( 12^2 \) and \( 10^2 \). This results in \( 144 \) and \( 100 \), respectively. Thus, the equation becomes: \[ c^2 = 144 + 100 - 2 \times 12 \times 10 \times \cos 40^{\circ} \].

Step 4: Compute the Cosine Component

Calculate \( 2 \times 12 \times 10 \), which gives \( 240 \). Multiply by \( \cos 40^{\circ} \). Use a calculator to find \( \cos 40^{\circ} \approx 0.7660 \). Then, \( 240 \times 0.7660 \approx 183.84 \).

Step 5: Complete the Calculation

Substitute the computed cosine component back: \[ c^2 = 144 + 100 - 183.84 \]. Simplifying gives \( c^2 = 244 - 183.84 = 60.16 \).

Step 6: Find the Final Result by Taking the Square Root

Take the square root of \( c^2 = 60.16 \) to solve for \( c \). This results in \( c \approx \sqrt{60.16} \approx 7.8 \) when rounded to the nearest tenth.

Key Concepts

Trigonometric Functions

Trigonometric functions are a central part of geometry and mathematics, used to relate angles to side lengths in right-angled and non-right-angled triangles. They include sine, cosine, and tangent. These functions are essential for solving problems that involve angles and lengths.

Cosine, specifically, is used in the Law of Cosines to determine unknown side lengths in triangles. It describes the ratio of the adjacent side to the hypotenuse in a right triangle. In non-right triangles, cosine helps relate an angle to the opposite and adjacent sides.

• This makes it invaluable for solving triangles with mixed types of angles and sides.
• In the Law of Cosines, cosine connects a known angle to the two known side lengths.

Understanding these trigonometric functions is crucial for navigating through triangular calculations efficiently!

Triangle Side Calculation

Triangle side calculation is a fundamental skill in geometry, allowing us to find unknown side lengths given certain known measurements. This particular exercise uses known side lengths and an angle to find the length of the unknown side.

The Law of Cosines is an extension of the Pythagorean theorem for triangles that aren't necessarily right-angled. It provides a method to calculate an unknown side when you have:

• Two side lengths
• The angle between those sides (the included angle)

The equation used is: \[ c^2 = a^2 + b^2 - 2ab \cos C \]In this formula, \( a \), \( b \), and \( C \) correspond to the two known sides and their included angle, making side calculation straightforward once plugged into this equation. This method enables us to handle complex triangle problems with greater ease and accuracy.

Cosine Rule

The Cosine Rule, also known as the Law of Cosines, is a powerful mathematical tool for calculating unknown sides or angles in triangles. It is vital for triangles that are not right-angled, where the Pythagorean theorem cannot be applied.

The Law of Cosines formula is:\[ c^2 = a^2 + b^2 - 2ab \cos C \]This formula shows the relation between all three sides of a triangle and one of its angles.

• The "\( c \)" represents the side we wish to calculate.
• \( a \) and \( b \) are the sides enclosing the known angle \( C \).

By knowing \( a \), \( b \), and \( C \), you can find the length of \( c \) accurately.The Cosine Rule comes into full effect especially in problems where you know two sides and the angle between them, providing an efficient solution to find the missing side. This rule is not only useful for side calculations but can also be adapted to find unknown angles, making it a flexible tool in trigonometry.