Problem 30 - Algebra and Trigonometry

Question

Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any resulting triangle(s). $$ a=2, \quad c=1, \quad A=120^{\circ} $$

Step-by-Step Solution

Answer

No triangles form since the given sides and angle do not satisfy the criteria to form any triangles.

Step 1: Identify the given values

The given values are: side a = 2, side c = 1, and angle A = 120 degrees.

Step 2: Determine the possible number of solutions using the Law of Sines

Use the Law of Sines to calculate side c: \ \[ \frac{a}{\text{sin}(A)} = \frac{c}{\text{sin}(C)} \ \frac{2}{\text{sin}(120^\textcirc)} = \frac{1}{\text{sin}(C)} \ \text{sin}(120^\textcirc) = \frac{\text{√3}}{2} \ \frac{2}{\frac{\text{√3}}{2}} = \frac{1}{\text{sin}(C)} \ \frac{4}{\text{√3}} = \frac{1}{\text{sin}(C)} \ \text{sin}(C) = \frac{\text{√3}}{4} \ \text{sin}(C) = 0.433 \ \text{Since } \text{C} = \text{sin}^{-1}(0.433), \text{there are no valid solutions since } 0.433 \text{ does not satisfy the sine inverse range within } [-1,1],\text. \text{This results in no triangle}. \]

Key Concepts

trigonometry

Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles. It is especially useful for solving problems involving triangles, whether they are right-angled or not. By understanding the core principles of trigonometry, you can solve a wide range of triangle-related problems using specific functions like sine, cosine, and tangent. These core functions help you connect the angles of a triangle with their corresponding side lengths.

triangle solutions

When solving triangles, knowing some key elements like sides and angles can help determine what kind of solution the triangle has. Here, the Law of Sines is crucial. It's especially helpful when dealing with oblique triangles, which are triangles that do not have a right angle. In our example, the given information includes two sides (a = 2, c = 1) and one angle (A = 120 degrees). Using these values, you can find out if you have one, two, or no triangle(s) possible.

angle and side relationships

One of the key elements in solving triangles is understanding the relationships between angles and sides. These relationships are typically governed by trigonometric rules and laws such as the Law of Sines and the Law of Cosines. In the given example, we use the Law of Sines, which says: \[\frac{a}{\text{sin}(A)} = \frac{c}{\text{sin}(C)} \]. Given angle A = 120 degrees, and sides a = 2 and c = 1, we solve for angle C using this relationship. However, in this case, the value of sine obtained (0.433) does not fall within the acceptable range, [-1,1], resulting in no valid triangle.

sine function

The sine function is fundamental in trigonometry as it relates the angle of a triangle to the ratio of the length of the opposite side to the hypotenuse. In the given exercise, the sine function helps us find unknown angles or sides. By using sin(A) and translating it into the ratio \(\frac{a}{c}\), you can solve for the unknowns according to the formula \(\text{sin}(A)\frac{a}{\text{sin}(A)} = \frac{c}{\text{sin}(C)}\). Here, the sine function showed that solving the equation results in a value not within the normal sine range, which implies that no valid triangle fits the given parameters.