Problem 34 - 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). $$ a=3, \quad b=7, \quad A=70^{\circ} $$

Step-by-Step Solution

Answer

No triangle can exist since \( \sin B \approx 2.195 \), which is greater than 1.

Step 1: Determine if the given information forms a triangle

To determine whether the given information results in one triangle, two triangles, or no triangle at all, use the Law of Sines: \[ \frac{a}{\sin A} = \frac{b}{\sin B} \]\ For the given values \( a = 3 \), \( b = 7 \), and \( A = 70^{\circ} \), solve for \( \sin B \).

Step 2: Solve for \( \sin B \)

Using the Law of Sines, express \( \sin B \): \[ \sin B = \frac{b \sin A}{a} \] Substitute the given values: \[ \sin B = \frac{7 \sin 70^{\circ}}{3} \] Evaluate this using the calculator: \[ \sin 70^{\circ} \approx 0.9397 \] \[ \sin B = \frac{7 \times 0.9397}{3} \approx 2.195 \] Since \( \sin B \) cannot be greater than 1, this implies no such triangle can exist.

Key Concepts

Law of Sines

The Law of Sines is a fundamental principle in trigonometry. It's particularly useful for solving triangles, especially when at least one side and its opposite angle are known. The Law of Sines states that the ratio of the length of a side of a triangle to the sine of its opposite angle is constant. Mathematically, it is expressed as:

• \frac{a}{ \frac{ \text{sin} \underline{\phantom{xxx}} A} } = \frac{ b}{ \frac{ \text{sin} \underline{\phantom{xxx}} B } } =\frac{ c }{ \frac{ \text{sin} \underline{\phantom{xxx}} C } }

This is true for any triangle, not just right-angled ones.

In the given exercise, we use this law to determine if a triangle can exist with the provided sides and angle. We need to solve for \(\text{sin} \underline{\phantom{xxx}} B\). By substituting the known values, if we find that \(\text{sin} \underline{\phantom{xxx}} B\) is greater than 1, then it indicates that no such triangle exists.

No triangle condition

Sometimes, when solving for a triangle using trigonometric relationships, you may find that no triangle is possible with the given parameters. This is known as the 'No Triangle Condition.' For instance, according to the Law of Sines, once we solved for \(\text{sin B}\) in our example, we obtained a value of approximately 2.195.

Since the sine of an angle in a real triangle must always be between -1 and 1 (inclusive), a value outside this range means that a corresponding angle cannot exist. Thus, in our exercise:

• If\( \text{sin} \underline{\phantom{xxx}} B \) exceeds 1 or is less than -1, it leads to the conclusion that no valid triangle can be formed with the given sides and angle.

This is critical for students to understand because it helps validate whether their geometric constructions are feasible.

Trigonometric functions

Trigonometric functions are essential tools in mathematics, specifically for solving triangles. The primary trigonometric functions are sine (sin), cosine (cos), and tangent (tan). In the context of our exercise, sine plays a crucial role.

When you need to determine an unknown side or angle of a triangle, these trigonometric functions come into play. For example, to determine \( \text{sin} \underline{\phantom{xxx}} A \) for a given angle A, we'd use a calculator or reference for values. In our solution:

• We found that\( \text{sin} \underline{\phantom{xxx}} 70^\text{o} \) approximates to 0.9397.

Using these values within the Law of Sines, students can explore the existences of triangles effectively.

Understanding trigonometric functions is vital not just in theory but also in practical applications such as physics, engineering, and even game development.