Problem 52 - Algebra 2

Question

Determine whether each triangle should be solved by beginning with the Law of Sines or Law of Cosines. Then solve each triangle. Round measures of sides to the nearest tenth and measures of angles to the nearest degree. $$ a=9, b=40, c=41 $$

Step-by-Step Solution

Answer

Use Law of Cosines first; angles: C=90°, A=13°, B=77°.

Step 1: Decide Which Law to Use

To determine whether to use the Law of Sines or the Law of Cosines, check the given values. We have all three sides of the triangle (a=9, b=40, and c=41) but none of the angles. In this case, the Law of Cosines is more appropriate because it can find an angle when all three sides are known.

Step 2: Use the Law of Cosines to Find Angle C

The Law of Cosines states that: \[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]Plugging in the known values,\[ 41^2 = 9^2 + 40^2 - 2(9)(40) \cdot \cos(C) \]\[ 1681 = 81 + 1600 - 720 \cdot \cos(C) \]Simplify to solve for \( \cos(C) \):\[ 1681 = 1681 - 720 \cdot \cos(C) \]\[ 0 = -720 \cdot \cos(C) \]Since the calculation 0 = 0 after simplification indicates correct process, finding \(\cos(C)\) means triangle is right-angled (angle \(C=90^\circ\)).

Step 3: Use the Law of Sines to Find Another Angle

Now that \(C=90^\circ\), use the Law of Sines to find angle \(A\):\[ \frac{\sin(A)}{a} = \frac{\sin(C)}{c} \]\[ \frac{\sin(A)}{9} = \frac{\sin(90^\circ)}{41} \]Since \(\sin(90^\circ) = 1\), solve for \(\sin(A)\):\[ \sin(A) = \frac{9}{41} \]Using a calculator, find \(A\) as:\(A \approx \arcsin\left(\frac{9}{41}\right)\) which approximately equals \(13^\circ\).

Step 4: Find the Remaining Angle

The sum of angles in a triangle is \(180^\circ\). Already knowing \(A\) and \(C\), solve for \(B\):\[ B = 180^\circ - 90^\circ - 13^\circ \]\[ B = 77^\circ \]

Key Concepts

triangle solving

When solving triangles, the objective is to find all its sides and angles using given values. For any triangle, we may have different sets of known values, such as:

• All three sides (SSS),
• Two sides and an angle (SAS or SSA),
• Two angles and a side (AAS or ASA).

Depending on the known values, different strategies can be used to solve the triangle, specifically using the Law of Sines or Cosines. For instance, if you know all three sides of a triangle (like in our exercise with sides 9, 40, and 41), the Law of Cosines is generally the best approach. This is because it allows you to calculate unknown angles directly from the sides. Essentially, the beginning of solving any triangle involves careful decision-making on which law or theorem will provide the most straightforward path to a solution.

Law of Sines

The Law of Sines is a powerful tool for solving triangles, especially useful when you know angles and sides in different configurations other than all three sides. The Law of Sines states:\[ \frac{\sin(A)}{a} = \frac{\sin(B)}{b} = \frac{\sin(C)}{c} \]This equation is extremely helpful when you need to find an unknown side or angle in a triangle where you know:

• Two angles and a side (AAS or ASA), or
• Two sides and a non-enclosed angle (SSA)

In our example, once we applied the Law of Cosines to establish the triangle was right-angled with angle C being 90 degrees, we then used the Law of Sines to find other angles. We knew side a and used the sine of angle C and side c. Knowing that \(\sin(90^\circ) = 1\), we rearranged the equation to solve for angle A. After determining angle A with the Law of Sines, we could easily find angle B using the sum of angles in a triangle.

trigonometric equations

Trigonometric equations are equations involving trigonometric functions like sine, cosine, and tangent. Solving these equations often involves finding angle measures that satisfy the equation, and they become essential in triangle solving when direct measurements are unavailable. In the given exercise, we started with a trigonometric equation: \[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]This equation, derived from the Law of Cosines, is a form of trigonometric equation. By substituting known numerical values of sides into this equation, we revealed information not just about the sides but about angle C. Trigonometric equations are not only used in isolated cases but often step into play together with the properties of triangles and key theorems, like in this exercise. Once the triangle's angle structure was partially revealed, even additional parts of the trigonometric toolkit such as the Law of Sines were utilized to further solve the triangle.