Problem 24 - Precalculus

Question

The measures of two sides and an angle are given. Determine whether a triangle (or two) exist, and if so, solve the triangle(s). $$\alpha=45^{\circ}, b=\sqrt{2}, a=1$$

Step-by-Step Solution

Answer

Only one triangle exists, an isosceles right triangle with angles 45°, 45°, and 90° and sides 1, 1, and \( \sqrt{2} \).

Step 1: Identify Type of Given Problem

This problem gives us two sides and a non-included angle. This situation corresponds to the ambiguous case of the Law of Sines, which can result in one triangle, two triangles, or no triangle.

Step 2: Apply Law of Sines to Check for Triangle Existence

Using the Law of Sines, \( \frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} \). Substitute the given values: \( \frac{1}{\sin(45^\circ)} = \frac{\sqrt{2}}{\sin(\beta)} \). This becomes \( \sqrt{2} \approx \frac{\sqrt{2}}{\sin(\beta)} \), implying \( \sin(\beta) = 1 \). Since \( \sin^{-1}(1) = 90^\circ \), angle \( \beta \) is 90°.

Step 3: Solve for the Remaining Angle

The angles in a triangle add up to 180°. With \( \alpha = 45^{\circ} \) and \( \beta = 90^{\circ} \), angle \( \gamma = 180^{\circ} - 45^{\circ} - 90^{\circ} = 45^{\circ} \).

Step 4: Find the Third Side Using Law of Sines

Using the Law of Sines again to find side \( c \): \( \frac{c}{\sin(\gamma)} = \frac{a}{\sin(\alpha)} \). With \( \gamma = 45^{\circ} \) and \( \alpha = 45^{\circ} \), \( \frac{c}{\sqrt{2}/2} = \frac{1}{\sqrt{2}/2} \), so \( c = 1 \).

Step 5: Conclusion

In this case, only one triangle exists, an isosceles right triangle with angles 45°, 45°, and 90° and sides 1, 1, and \( \sqrt{2} \).

Key Concepts

Ambiguous Case

The ambiguous case arises in trigonometry when you're given two sides and a non-included angle of a triangle. This situation is particularly linked to the Law of Sines and often results in uncertainty about the number of triangles that could be formed.

In the exercise given, the ambiguous case is recognized because you know two side lengths, \( a \) and \( b \), and an angle \( \alpha \), without knowing if it is included between these sides. Using the Law of Sines, you apply the formula \( \frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} \) to find if an angle \( \beta \) exists that satisfies the condition, thus defining a valid triangle.

In many cases, the ambiguous nature means:

• One valid triangle exists.
• Two distinct triangles exist.
• No triangle at all can exist given those measurements.

Special attention to this case is important to avoid mistakes, particularly in determining if the angles are correct and consistent with triangular geometry.

Triangle Existence

Triangle existence is determined by the possibility of forming a triangle given some measurements pertaining to its angles and sides. Key mathematical principles like the Law of Sines and trigonometric properties help define if triangles can exist.

In this problem, checking for triangle existence requires establishing the possibility of an angle (\( \beta \)) such that \( \sin(\beta) = 1 \), which tells us that \( \beta = 90^{\circ} \).

Once you ascertain \( \beta \), verifying the sum of angles within the triangle becomes the next step:

• Sum of angles must be \( 180^{\circ} \)
• Your given \( \alpha = 45^{\circ} \) and \( \beta = 90^{\circ} \) automatically define \( \gamma = 45^{\circ} \)

It systematically confirms that this set of parameters can indeed exist as a triangle. However, if the angle sum doesn’t reach \( 180^{\circ} \), then a triangle cannot exist, confirming one of the aforementioned three scenarios.

Isosceles Right Triangle

An isosceles right triangle is a special type of right triangle where the two legs are congruent, meaning they have the same length. This triangle features two angles at \( 45^{\circ} \) and one right angle at \( 90^{\circ} \).

Having solved the problem, it's clear that an isosceles right triangle is the only possible solution in this scenario:

• Each of the angles apart from the right angle is \( 45^{\circ} \).
• The sides opposite these angles are equal, both measuring \( 1 \) in this solution.

The hypotenuse in an isosceles triangles, such as this, is \( \sqrt{2} \). This precise and configured property helps simplify many calculative scenarios, reinforcing the importance of verifying measurements, angles, and using trigonometric laws thoroughly. Such triangles are foundational in geometric studies and help create a strong understanding of angular relationships and symmetry.