Problem 12 - Precalculus

Question

Use degree measure for your answers. In parts (c) and (d), use a calculator and round the results to one decimal place. (a) Show that there is no triangle with \(a=2, b=3,\) and \(\angle A=42^{\circ}\) (b) Is there any triangle in which \(a=2, b=3,\) and \(A=41^{\circ} ?\)

Step-by-Step Solution

Answer

(a) No such triangle exists. (b) Yes, a triangle can exist.

Step 1: Analyze the given triangle for part (a)

We are given a triangle where side \(a = 2\), side \(b = 3\), and angle \(\angle A = 42^\circ\). To determine if this triangle exists, we need to assess if the Law of Sines and the given sides and angle allow the construction of a triangle. We'll calculate \(\sin B\) by rearranging the Law of Sines formula: \[ \frac{a}{\sin A} = \frac{b}{\sin B} \] Substituting the known values, we get:\[ \frac{2}{\sin 42^\circ} = \frac{3}{\sin B} \] Solving for \(\sin B\), we find: \[ \sin B = \frac{3 \cdot \sin 42^\circ}{2} \].

Step 2: Calculation for part (a)

Calculate \(\sin 42^\circ\) using a calculator:\[ \sin 42^\circ \approx 0.6691 \] Then, substitute back into the equation:\[ \sin B = \frac{3 \times 0.6691}{2} = 1.00365 \] Since \(\sin B\) must be between -1 and 1 for a valid angle, this indicates there is no possible angle \(B\) fulfilling these conditions. Therefore, no such triangle exists.

Step 3: Analyze the given triangle for part (b)

Now, let's consider the triangle with \(a = 2\), \(b = 3\), and \(\angle A = 41^\circ\). Using the Law of Sines again:\[ \frac{a}{\sin A} = \frac{b}{\sin B} \] Substitute the given values:\[ \frac{2}{\sin 41^\circ} = \frac{3}{\sin B} \] Solve for \(\sin B\): \[ \sin B = \frac{3 \cdot \sin 41^\circ}{2} \].

Step 4: Calculation for part (b)

Calculate \(\sin 41^\circ\) using a calculator:\[ \sin 41^\circ \approx 0.6561 \] Substitute back:\[ \sin B = \frac{3 \times 0.6561}{2} = 0.98415 \] Since \(\sin B\) is between -1 and 1, \(B\) is valid. We can find \(B\) using \(\sin^{-1}(0.98415)\), and complete the triangle using \[\angle C = 180^\circ - 41^\circ - B\]. This confirms that a triangle with these dimensions can exist.

Key Concepts

Law of Sines

The Law of Sines is a helpful tool in trigonometry that connects the lengths of sides of a triangle with the sines of its angles. It's used to solve for unknown angles or sides in non-right triangles. The formula is expressed as:

• \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \)

This means that the ratio of the length of a side to the sine of its opposite angle is constant for all three sides of the triangle.

The Law of Sines is particularly useful when we have:

• Two angles and one side (AAS or ASA situation).
• Two sides and a non-included angle (SSA scenario, like in our problem).

Degree Measure

Angles can be measured in several ways, but the most common unit is degrees. A full circle is divided into 360 equal parts, called degrees.

In trigonometry, degree measure helps in easily calculating angles with tools like calculators which usually have a degree input mode. It's important to ensure your calculator is in degree mode when working with angles in degrees.

• For example, using degree measure, we can state that \( \sin 41^\circ \approx 0.6561 \).
• It simplifies representation and understanding of angle sizes compared to radians or other units.

Triangle Existence

Not every set of side lengths and angles can form a triangle. To check if a triangle is possible, we use certain rules. With the given data in a problem, it's necessary to verify these conditions relevant to the problem:

• The sum of the angles must be \( 180^\circ \).
• The sum of any two sides must be greater than the third side (Triangle Inequality Theorem).

For our specific problem, we used the Law of Sines to find whether an angle was feasible:

• If \( \sin B > 1 \) or \( \sin B < -1 \), then no such angle \( B \) that fulfills the Law of Sines exists, meaning the triangle cannot be formed.
• In part (a), \( \sin B = 1.00365 \) which is invalid, so no triangle can exist.
• In part (b), \( \sin B = 0.98415 \) which is valid, so a triangle can be formed.

Angle Calculation

To find angles in a triangle when we have at least one side length and an opposite angle, we utilize trigonometric functions such as sine.

Consider these steps when using the Law of Sines for angle calculations:

• Solve for the unknown sine of angle using the rearranged Law of Sines formula.
• Use a calculator to find the sine value for an angle, ensuring your calculations are in degree mode.
• Apply the inverse sine function (\( \sin^{-1} \)) to calculate the angle's measure from a known sine value.

When calculating, remember to round angles to the specified precision, typically to one decimal place, to ensure accuracy in your solutions.