Problem 14 - Precalculus with Limits

Question

Using the Law of Sines. Use the Law of Sines to solve the triangle. Round your answers to two decimal places. $$A=120^{\circ}, \quad B=45^{\circ}, \quad c=16$$

Step-by-Step Solution

Answer

The third angle C is \(15^{\circ}\). The sides 'a' and 'b' can be found using the Law of Sines and the provided information. The final values of 'a' and 'b' need to be rounded off to two decimal places for the final answer.

Step 1: Calculate the third angle

Given angles A and B, we can calculate angle C by using the property that sum of angles in a triangle is \(180^{\circ}\). So, \(C = 180^{\circ} - A - B = 180^{\circ} - 120^{\circ} - 45^{\circ} = 15^{\circ}\)

Step 2: Using the Law of Sines

Now, we use the Law of Sines 'The ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides and angles.' to find the remaining sides a and b. \(\frac{a}{\sin(A)} = \frac{c}{\sin(C)}\). Solving this, we get \(a = c * \frac{\sin(A)}{\sin(C)} = 16 * \frac{\sin(120)}{\sin(15)}\)

Step 3: Calculate side b

Similarly, we can calculate side b using the Law of Sines. \(\frac{b}{\sin(B)} = \frac{c}{\sin(C)}\). Solving for b, we get \(b = c * \frac{\sin(B)}{\sin(C)} = 16 * \frac{\sin(45)}{\sin(15)}\)

Step 4: Round off the answers

Now, we round off the calculated values of sides a and b to two decimal places. So, for a value of x.xxxxx, rounding off gives us x.xx. Similarly do it for a and b.

Key Concepts

Solving Triangles

The process of finding all the unknown sides and angles of a triangle is known as solving triangles. This is a fundamental task in trigonometry. When we have a triangle with some known angles and sides, we can use various methods to find the remaining angles and sides.

For example, when we have two angles and one side, like in the given exercise with angles A and B and side c, we can find the third angle easily because the angles in a triangle always add up to 180 degrees. Once we have all the angles, we can use the Law of Sines or the Law of Cosines to find the unknown sides. Solving triangles allows us to tackle a multitude of problems in different fields like engineering, astronomy, and navigation.

Trigonometric Ratios

Trigonometric ratios are the ratios between the sides of a right triangle and the angles within the triangle. The primary trigonometric ratios are sine (sin), cosine (cos), and tangent (tan), and they depend on a specific angle of the triangle.

The Law of Sines relates these ratios to the sides of any (not only right) triangle. In our exercise, we're using the sine ratio, which is the ratio of the length of the opposite side to the hypotenuse for a right triangle, but in the context of the Law of Sines, it is used to establish a relationship between all sides of a triangle and their opposite angles. By knowing one side and its opposite angle, the Law of Sines allows us to find the unknown sides.

Angle Sum Property

The angle sum property of a triangle simply states that the sum of the angles in a triangle always equals 180 degrees. In mathematical terms: if the angles are named A, B, and C, then the rule is represented as \( A + B + C = 180^\circ \).

This property is invaluable when solving triangles because if we know any two angles, we can always find the third. In our current exercise, knowing that angles A and B are 120 degrees and 45 degrees respectively, we used this property to deduce that angle C must be 15 degrees to ensure the sum is 180 degrees.

Rounding Decimal Places

In mathematics, and specifically in numerical calculations, rounding decimal places is the process of adjusting a number to be approximately equal to a certain precision. When rounding to two decimal places, you look at the third decimal place. If this digit is 5 or higher, we round the second decimal up by one; if it is 4 or lower, we leave the second decimal as it is.

In the given exercise, after calculating sides a and b, we must round our answers to two decimal places. This is often done to simplify the numbers for ease of communication or when the exact decimal is not required for practical purposes. Rounding also helps to address the issue of significant figures and the limit of accuracy of the measurements.