Problem 14 - Algebra 2 and Trigonometry

Question

In \(3-14 :\) a. Determine the number of possible triangles for each set of given measures. b. Find the measures of the three angles of each possible triangle. Express approximate values to the nearest degree $$ a=8, b=10, \mathrm{m} \angle A=45 $$

Step-by-Step Solution

Answer

One triangle is possible with angles approximately \(45^\circ\), \(62^\circ\), and \(73^\circ\).

Step 1: Determine Type of Triangle

Apply the law of sines to determine if a triangle can be formed. Given, \(a = 8\), \(b = 10\), \(\angle A = 45^\circ\). We set up the law of sines equation: \(\frac{a}{\sin A} = \frac{b}{\sin B}\). Solving for \(\sin B\), we have \(\sin B = \frac{b \cdot \sin A}{a} = \frac{10 \cdot \sin 45^\circ}{8}\). Calculate \(\sin B\) to determine if there's a valid value between 0 and 1.

Step 2: Calculate \(\sin B\)

Calculate \(\sin 45^\circ\), which is approximately \(0.7071\). Substitute into the equation: \(\sin B = \frac{10 \times 0.7071}{8} = \frac{7.071}{8} \approx 0.8839\). This value is between 0 and 1, confirming a valid \(\angle B\).

Step 3: Compute \(\angle B\)

Use the inverse sine function: \(\angle B = \sin^{-1}(0.8839)\). Compute this angle to find \(\angle B \approx 62^\circ\).

Step 4: Compute \(\angle C\)

Use the fact that the sum of angles in a triangle is \(180^\circ\). Thus, \(\angle C = 180^\circ - \angle A - \angle B = 180^\circ - 45^\circ - 62^\circ = 73^\circ\).

Step 5: Conclusion

There is only one possible triangle with the given parameters, and its angles are approximately \(45^\circ\), \(62^\circ\), and \(73^\circ\).

Key Concepts

Law of Sines

The Law of Sines is a handy tool in trigonometry when dealing with non-right triangles. It relates the lengths of the sides of a triangle to the sines of its angles. This law is crucial for situations where you have partial knowledge of a triangle’s side lengths and angles, like we have here with the measures given: side lengths and one angle.

To use the Law of Sines, you apply the formula:

• \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \)

Here, \(a\), \(b\), and \(c\) are the sides of the triangle opposite the angles \(A\), \(B\), and \(C\), respectively. Using the known angle (\(\angle A = 45^\circ\)) and side lengths (\(a = 8\) and \(b = 10\)), we can find the sine of other angles within the triangle.

As shown in the solution, to find \(\sin B\), substitute the values into the Law of Sines equation, which gives you \( \sin B = \frac{10 \times 0.7071}{8} \). This setup helps determine if the triangle is valid by checking if the calculated sine values are between 0 and 1.

Angle Calculation

Calculating angles in a triangle involves using both trigonometric values and inverse functions. Once you have computed a sine value like \(\sin B = 0.8839\), obtaining the actual angle requires the inverse sine function (also known as \(\sin^{-1}\) or arcsin). This process is often aided by calculators that have trigonometric functions.

In our exercise, once \(\sin B\) is known, the next step is to find \(\angle B\) using:

• \( \angle B = \sin^{-1}(0.8839) \)

The arcsin value here tells us that \(\angle B\) is approximately \(62^\circ\). This reveals the measure of the second angle in the triangle.
Understanding how to use the inverse functions is vital because they allow us to convert the calculated trigonometric values back into angle measures.

Triangle Angle Sum

One of geometry's fundamental rules is that the sum of the internal angles of a triangle is always \(180^\circ\). This principle is critical in validating the obtained measures of a triangle's angles.

In the given solution, after determining that \(\angle A = 45^\circ\) and \(\angle B \approx 62^\circ\), the last piece was to calculate \(\angle C\) using the triangle angle sum rule. This is done by subtracting the sum of the known angles from \(180^\circ\).

• \( \angle C = 180^\circ - \angle A - \angle B = 180^\circ - 45^\circ - 62^\circ \)
• This calculation gives \(\angle C \approx 73^\circ\)

Applying this rule ensures completeness and accuracy in triangle measurements, confirming the triangle's angles indeed total \(180^\circ\). This not only checks the correctness of individual calculations but also assures the logical consistency of the entire problem-solving approach.