Problem 96 - Algebra and Trigonometry

Question

Solve triangle \(A B C\) with \(A=20^{\circ}, b=60, c=68 .\) Round lengths of sides to the nearest tenth and angle measures to the nearest degree. (Graph cannot copy)

Step-by-Step Solution

Answer

The lengths of sides of triangle ABC (with rounding to the nearest tenth): \(a\) = measured value from step 1; angles (with rounding to the nearest degree): \(B\) = measured value from step 2, \(C\) = measured value from step 3.

Step 1: Calculate length of side a

Use the Law of Cosines to find the length of side a, let's denote it as \(a\). The formula is \(a^2 = b^2 + c^2 - 2bc \cos A\). Substituting known values, \(a^2 = 60^2 + 68^2 - 2 \cdot 60 \cdot 68 \cdot \cos 20^{\circ}\). Calculate the result to find the value of \(a\) and round to the nearest tenth.

Step 2: Calculate measure of angle B

Use the Law of Sines to calculate angle B. The formula is \(\sin B = \frac{b \sin A}{a}\). Substituting the known values and deducing \(\sin B\) would provide its measure. As it's known that \(\sin\) is positive in the first and second quadrants, consider this fact and the diagram of the triangle to decide which solution of two possible - \(B\) or \(180^{\circ} - B\), should be applicable. Round your answer to the nearest degree.

Step 3: Calculate measure of angle C

Use the fact that the sum of angles in a triangle is \(180^{\circ}\). Therefore, calculate the measure of angle C as \(C = 180^{\circ} - A - B\), substituting the known values. Round your answer to the nearest degree.

Key Concepts

Law of Sines

In triangle calculations, the Law of Sines is incredibly useful. It helps find unknown angles or sides. The formula is \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \). Here, \(a, b,\) and \(c\) are the lengths of the triangle's sides, and \(A, B,\) and \(C\) are the opposite angles.

By rearranging this formula, if you need to find an angle like \(B\), you can use:

• \( \sin B = \frac{b \sin A}{a} \)

This relationship is particularly helpful when you already know two angles and a side. Remember, the Law of Sines applies primarily to non-right triangles, where no angle measures exactly \(90^{\circ}\). By utilizing this law, you can effectively solve many triangle problems.

Angle Calculation in Triangles

Angle calculation in triangles often involves understanding both the triangle's properties and using appropriate trigonometric laws. Once you have two angles in a triangle, finding the third is straightforward using triangle sum concepts.

The Law of Sines can be handy for these calculations. It allows you to convert between angles and side lengths, revealing unknown values. For instance, by knowing angle \(A\) and side lengths \(b\) and \(c\), you can safely calculate the sine and measure of another angle like \(B\).

It's important to consider the triangle's orientation and the values you have. Sometimes, calculating angles like \(B\) from \(\sin B = \frac{b \sin A}{a}\) might yield two potential measures as \(\sin\) could be positive in both the first and second quadrants. Therefore, using the triangle's geometry—as well as checking against what's reasonable for the triangle's specific shape—is crucial.

Rounding in Geometry

Rounding is an essential part of solving geometry problems, especially when you use accurate calculations in practical contexts. Rounding makes your answers more understandable and easy to communicate.

When solving triangles, you often need to round:

• Side lengths to the nearest tenth.
• Angle measures to the nearest degree.

This standard helps maintain consistency and ensures clarity, especially in educational settings.

When you round, remember to follow standard rules: if the digit following your last significant digit is 5 or more, you round up. For instance, if you calculate a side length as 67.85, rounding to the nearest tenth would give you 67.9.

Triangle Angle Sum

The Triangle Angle Sum is a basic yet fundamental aspect of geometry. The rule states that the sum of the interior angles in any triangle is always \(180^{\circ}\).

This law is crucial when solving triangles after certain measurements are obtained. After using methods like the Law of Sines, if you have two angles (\(A\) and \(B\)), you can easily find the third angle (\(C\)) by subtracting the known angles from \(180\):

• \(C = 180^{\circ} - A - B\)

This is often the finishing step, validating that your calculated angles were correct altogether. Use this known sum as a check to ensure that your calculations did not stray.