Problem 11 - Algebra 2 and Trigonometry

Question

In \(7-12,\) find the cosine of each angle of the given triangle. In \(\triangle M N P, m=16, n=15, p=8\)

Step-by-Step Solution

Answer

\(\cos(M) \approx 0.1375\), \(\cos(N) \approx 0.3711\), \(\cos(P) \approx 0.8688\).

Step 1: Understanding the Given Triangle

The problem provides the sides of the triangle \(\triangle MNP\) where \(m=16\), \(n=15\), and \(p=8\). Here \(m\), \(n\), and \(p\) represent the lengths of the sides opposite to angles \(M\), \(N\), and \(P\) respectively.

Step 2: Apply the Law of Cosines to Find \(\cos(M)\)

To find \(\cos(M)\), we use the Law of Cosines: \[ c^2 = a^2 + b^2 - 2ab \cos(C) \]For angle \(M\), substituting \(a=15\), \(b=8\), and \(c=16\):\[ 16^2 = 15^2 + 8^2 - 2\cdot15\cdot8 \cdot \cos(M) \]\[ 256 = 225 + 64 - 240 \cos(M) \]\[ 256 = 289 - 240 \cos(M) \]Rearranging gives:\[ 240 \cos(M) = 289 - 256 \]\[ 240 \cos(M) = 33 \]\[ \cos(M) = \frac{33}{240} \approx 0.1375 \]

Step 3: Apply the Law of Cosines to Find \(\cos(N)\)

Now we use the Law of Cosines for \(\cos(N)\):\[ n^2 = m^2 + p^2 - 2mp \cos(N) \]Substituting \(m=16\), \(n=15\), and \(p=8\):\[ 15^2 = 16^2 + 8^2 - 2\cdot16\cdot8 \cdot \cos(N) \]\[ 225 = 256 + 64 - 256 \cos(N) \]\[ 225 = 320 - 256 \cos(N) \]Rearranging gives:\[ 256 \cos(N) = 320 - 225 \]\[ 256 \cos(N) = 95 \]\[ \cos(N) = \frac{95}{256} \approx 0.3711 \]

Step 4: Apply the Law of Cosines to Find \(\cos(P)\)

Finally, find \(\cos(P)\) using the Law of Cosines:\[ p^2 = m^2 + n^2 - 2mn \cos(P) \]Substitute \(m=16\), \(n=15\), and \(p=8\):\[ 8^2 = 16^2 + 15^2 - 2\cdot16\cdot15 \cdot \cos(P) \]\[ 64 = 256 + 225 - 480 \cos(P) \]\[ 64 = 481 - 480 \cos(P) \]Rearranging gives:\[ 480 \cos(P) = 481 - 64 \]\[ 480 \cos(P) = 417 \]\[ \cos(P) = \frac{417}{480} \approx 0.8688 \]

Key Concepts

Cosine of an Angle

The cosine of an angle in a triangle is an important concept in trigonometry as it helps in determining unknown angles or sides in a triangle. It is represented by the ratio of the adjacent side to the hypotenuse in a right-angled triangle. However, for any triangle, we can use the Law of Cosines to find the cosine of a given angle.
In a triangle, for any angle \( C \), if we know the lengths of all three sides \( a \), \( b \), and \( c \), the cosine of angle \( C \) can be calculated using the formula:
\[ c^2 = a^2 + b^2 - 2ab \cos(C) \]This formula is useful in various scenarios such as in navigation, architecture, and physics, where measuring angles directly may be difficult. Understanding the cosine helps you not only find angles but also solve real-world problems that involve triangular shapes.

Triangle Side Lengths

Knowing the lengths of the sides of a triangle is crucial for calculating angles using trigonometric functions like cosine. Each side of a triangle has a specific name when related to its opposing angle:

• \( a \), \( b \), and \( c \) for the sides opposite to angles \( A \), \( B \), and \( C \) respectively.
• These lengths provide a basis to apply the Law of Cosines effectively to determine unknown angles or sides.

When given all three side lengths of a triangle, you can utilize these lengths to find angles by rearranging and applying the Law of Cosines:
\[ c^2 = a^2 + b^2 - 2ab \cos(C) \]
This is vital for solving many problems in mathematics and allows us to explore the properties of triangles beyond simple right-angled forms, expanding our understanding and ability to solve complex geometrical problems.

Trigonometry

Trigonometry focuses on the relationships between the angles and sides of triangles, making it a powerful tool in both pure and applied mathematics. Originating from the Greek words "trigonon" (triangle) and "metron" (measure), it allows for calculating unknown elements of a triangle when some are known.
There are six primary trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant. The cosine function specifically deals with the adjacent side and hypotenuse in right triangles, expanding to the Law of Cosines in general triangles.
Trigonometry is applied in various fields such as:

• Architecture and engineering for designing structures and analyzing forces.
• Physics for understanding wave functions, oscillations, and vectors.
• Navigation and geography to determine locations and distances using bearings and coordinates.

Mastering the basics of trigonometry, including the Law of Cosines, enables solving diverse and complex problems efficiently.