Problem 26 - Precalculus

Question

Use the given information to find the cosine of each angle in \(\triangle \overline{A B C}\) \(a=17 \mathrm{cm}, b=8 \mathrm{cm}, c=15 \mathrm{cm}\) (For this particular tri- angle, you can check your answers, because there is an alternative method of solution that does not require the law of cosines.)

Step-by-Step Solution

Answer

\(\cos A = 0\), \(\cos B = \frac{15}{17}\), \(\cos C = \frac{8}{17}\).

Step 1: Applying the Law of Cosines to Find \(\cos A\)

The Law of Cosines is given by \[\cos A = \frac{b^2 + c^2 - a^2}{2bc}\.\] Substitute in the given values: \(a = 17\) cm, \(b = 8\) cm, \(c = 15\) cm to find \(\cos A\). \[ \cos A = \frac{8^2 + 15^2 - 17^2}{2 \times 8 \times 15} = \frac{64 + 225 - 289}{240} = \frac{0}{240} = 0.\] Therefore, \(\cos A = 0\).

Step 2: Applying the Law of Cosines to Find \(\cos B\)

Now apply the Law of Cosines to find \(\cos B\): \[\cos B = \frac{a^2 + c^2 - b^2}{2ac}\.\] Substitute in the given values: \(a = 17\) cm, \(b = 8\) cm, \(c = 15\) cm.\[ \cos B = \frac{17^2 + 15^2 - 8^2}{2 \times 17 \times 15} = \frac{289 + 225 - 64}{510} = \frac{450}{510} = \frac{15}{17}\.\] Thus, \(\cos B = \frac{15}{17}\).

Step 3: Applying the Law of Cosines to Find \(\cos C\)

Use the Law of Cosines to find \(\cos C\): \[\cos C = \frac{a^2 + b^2 - c^2}{2ab}\.\] Substitute the given values \(a = 17\) cm, \(b = 8\) cm, \(c = 15\) cm.\[ \cos C = \frac{17^2 + 8^2 - 15^2}{2 \times 17 \times 8} = \frac{289 + 64 - 225}{272} = \frac{128}{272} = \frac{16}{34} = \frac{8}{17}.\] Thus, \(\cos C = \frac{8}{17}\).

Key Concepts

Law of Cosines

The Law of Cosines is an essential tool in trigonometry for solving triangles when you know:- Two sides and a non-enclosed angle- Three sidesThis law helps us find the unknown angles in triangles. It is invaluable when dealing with problems not easily addressed by simpler trigonometric functions like sine and tangent. The formula can be summarized for any angle in a triangle as:\[\cos A = \frac{b^2 + c^2 - a^2}{2bc}\]where \(a\), \(b\), and \(c\) are the lengths of the sides of the triangle.
The formula is essentially an extension of the Pythagorean theorem. It adjusts based on the angles and side lengths of the non-right triangle. This means that, similar to the Pythagorean theorem, the Law of Cosines is about the relationship between angles and sides. Using this law is straightforward once values are substituted, as seen in the solved example where substituting the side lengths into the formula yields the cosines of angles \(A\), \(B\), and \(C\). Applying these formulas correctly allows you to solve various triangle-related problems efficiently.

triangle problem solving

Solving triangle problems often involves figuring out measurements like angles or side lengths. When given specific values like side lengths in the exercise, understanding the relationships between them is key. In triangle problem-solving:

• Identify known values and what needs to be found.
• Choose the appropriate formulas or theorems to apply.
• Substitute known values into these formulas to find unknowns.

During the solution steps, each angle's cosine was found using its respective expression via the Law of Cosines. This involves substituting the triangle's sides directly into the formula to solve for each angle's cosine. It's quite similar to piecing together a puzzle where each correct formula-part substitution helps find another missing piece – in this case, the angle.
It's crucial to ensure calculations are accurate, as small mistakes can lead to incorrect solutions. This methodical approach helps build a solid understanding of the triangle's properties and relationships.

trigonometry

Trigonometry is the mathematics of triangles, particularly right-angled ones, but also applicable to any type of triangle. It's built around understanding the relationships between angles and sides. Trigonometric concepts, like cosine, sine, and tangent, provide methods to analyze these relationships deeply. The cosine, specifically, measures the adjacent side of a right triangle over its hypotenuse, extended in the Law of Cosines to work beyond right triangles.
In everyday triangle problems, trigonometry helps decode scenarios by providing functional relationships. The angle found via the cosine can further lead to calculations of other trigonometric functions. This highlights how one trigonometric function can serve as a stepping stone to fully solve or understand a triangle's properties. Understanding trigonometry allows students to solve real-world problems, such as calculating heights or distances indirectly. The exercise about finding cosines of angles underscored this principle as it expanded the utility of trigonometry beyond basics via the sophisticated Law of Cosines. This makes trigonometry essential for many scientific and engineering applications.