Problem 16 - Algebra and Trigonometry

Question

In Exercises 9–16, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. $$ B=80^{\circ}, C=10^{\circ}, a=8 $$

Step-by-Step Solution

Answer

The triangle measures are: Angle A = 90 degrees, Side b = 7.9 units, Side c = 1.4 units.

Step 1: Calculate remaining angle A

Subtract the sum of angles B and C from 180. Angle A = 180 - (80 + 10) = 90 degrees.

Step 2: Use Law of Sines to find side b

The law of sines states that the ratio of a side length to the sine of its opposite angle is the same for all three sides of the triangle. Therefore, we can write: \[ b = \frac{a}{\sin A}\times{\sin B} = \frac{8}{\sin 90}\times{\sin 80} = 7.9\] (rounded to the nearest tenth)

Step 3: Use Law of Sines to find side c

Similarly, use the law of sines to find side c: \[ c = \frac{a}{\sin A}\times{\sin C} = \frac{8}{\sin 90}\times{\sin 10} = 1.4\] (rounded to the nearest tenth).

Key Concepts

Law of Sines

The Law of Sines is an essential principle for solving triangles, especially when dealing with non-right-angled cases. It states that the ratio of a side length of a triangle to the sine of its opposite angle is constant for all three sides and angles within the triangle. In mathematical terms, for a triangle with angles A, B, C and opposite sides a, b, c, the Law of Sines is written as:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]
This is particularly useful when you have a triangle where you know one side length and its opposite angle, along with another angle or side. It allows you to solve for the unknown sides or angles. For example, if we know angle B, angle C, and side a, as in the given exercise, we can find side b by rearranging the formula:
\[ b = \frac{a}{\sin A} \times \sin B \]
And similarly for side c. Understanding this law is crucial for solving triangles as it provides a direct method to find missing measurements.

Angle Measurement

Angle measurement refers to determining the size of an angle in degrees, radians, or other units. Triangles have a total of 180 degrees, and knowing this allows us to calculate missing angle measures when other angles are known. For instance, in the exercise, angles B and C are given, from which angle A can be worked out by subtracting the sum of these angles from 180 degrees. Accurate angle measurement is indispensable as it directly relates to applying the Law of Sines.

When solving triangles, always ensure that the sum of your angle measurements is 180 degrees to validate the correctness of your calculation. Measuring angles with precision is key for correctly applying laws of trigonometry, as even a small error can significantly affect the calculated side lengths.

Side Length Calculation

Calculating side lengths in a triangle involves determining the lengths of unknown sides based on the triangle's known angles and sides. As seen in the provided exercise, once we have an angle measure, we can use trigonometric laws, such as the Law of Sines, to calculate unknown side lengths. The process typically involves setting up a proportion using the known angle and opposite side, and solving for the missing side length.

It is crucial to round the results appropriately, as the problem often specifies the degree of precision required. Round lengths to the nearest tenth for an exercise like this one. Precise calculation and rounding are essential to ensure that the dimensions are practical for application in real-life scenarios or further mathematical problems.