Problem 23 - Precalculus Mathematics for Calculus

Question

Use the Law of sines to solve for all possible triangles that satisfy the given conditions. $$b=25, \quad c=30, \quad \angle B=25^{\circ}$$

Step-by-Step Solution

Answer

The valid triangle has angles: \(A \approx 124.52^\circ\), \(B = 25^\circ\), \(C \approx 30.48^\circ\).

Step 1: Understand the Law of Sines

The Law of Sines states that for any triangle with sides labeled \(a\), \(b\), and \(c\), and their opposite angles labeled \(A\), \(B\), and \(C\) respectively, the following ratio holds: \[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}. \] We'll use this to find the angles and sides of the given triangle.

Step 2: Use the Law of Sines to find angle C

Apply the Law of Sines to find angle \(C\) using the given parameters. We have:\[ \frac{c}{\sin C} = \frac{b}{\sin B}, \]which rearranges to: \[ \sin C = \frac{c \cdot \sin B}{b} = \frac{30 \cdot \sin 25^\circ}{25}. \]Calculate \(\sin C\): \[ \sin C \approx \frac{30 \cdot 0.4226}{25} \approx 0.5071. \]

Step 3: Determine possible angle C values

Since \(\sin C = 0.5071\), based on the sine function property, we can have two possible angles \(C\): \(C_1 = \arcsin(0.5071) \approx 30.48^\circ \) and \(C_2 = 180^\circ - 30.48^\circ \approx 149.52^\circ\).

Step 4: Calculate angle A for both scenarios

For \(C_1 = 30.48^\circ\), calculate \(A_1\): \(A_1 = 180^\circ - B - C_1 = 180^\circ - 25^\circ - 30.48^\circ \approx 124.52^\circ\).For \(C_2 = 149.52^\circ\), calculate \(A_2\): \(A_2 = 180^\circ - B - C_2 = 180^\circ - 25^\circ - 149.52^\circ = 5.48^\circ\).Check feasibility: Only \(C_1 = 30.48^\circ\) forms a valid triangle as all angles are positive and sum to 180 degrees.

Step 5: Solve for side a using angle A_1

Using the Law of Sines again, find side \(a\) given \(A_1\):\[ \frac{a}{\sin A_1} = \frac{b}{\sin B}, \]which rearranges to: \[ a = \sin A_1 \cdot \frac{b}{\sin B} = \sin 124.52^\circ \cdot \frac{25}{0.4226}. \]Calculate \(a\):\(a \approx 0.8231 \cdot \frac{25}{0.4226} \approx 48.65.\)

Key Concepts

Triangle Solutions

Solving a triangle involves finding all its unknown sides and angles, given some initial information. This can be done using different trigonometric rules, such as the Law of Sines.
In the exercise above, we are given two sides, \(b\) and \(c\), and an angle \(B\). This is a typical "ASA" (angle-side-angle) or "SSA" (side-side-angle) situation, where you have enough information to explore the triangle further using trigonometric principles.

• First, the Law of Sines is used to find another angle or side.
• Once you have this information, the triangle can usually be solved completely.

Remember, the sum of the angles in any triangle must equal \(180^\circ\). This fundamental rule allows you to calculate any missing angles once you have two of them. The process often involves iterating calculations to find the most accurate triangle description.

Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between side lengths and angles of triangles. In the context of triangle solutions, trigonometry provides powerful tools like the Law of Sines and Cosines.
These laws help mathematicians and students to calculate unknown parts of a triangle when certain combinations of sides and angles are known.

• The Law of Sines states: \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \).
• This relationship is incredibly useful when dealing with non-right angled triangles, such as the one in the exercise.

Using calculators to find sine values and understanding how sine relates to triangle sides and angles are necessary skills.
With these tools, solving triangles becomes a step-by-step process of applying trigonometric identities to find missing values.

Ambiguous Case

The "ambiguous case" in trigonometry occurs when applying the Law of Sines to solve a triangle creates two possible answers for an angle. This typically happens in an "SSA" situation, where two sides and a non-included angle are known.

• In the exercise, this ambiguity is seen with angle \(C\), where two possible values, \(C_1\) and \(C_2\), are initially considered.
• The angle can either be \( \arcsin(\sin C)\) or \(180^\circ - \arcsin(\sin C)\), as the sine function is positive in the first and second quadrants.

However, not every calculated triangle is feasible.
This demands checking whether the triangle's angles add up to \(180^\circ\) and all angles are positive.
Once filtered, only valid triangles are considered for final solutions, removing the ambiguity effectively.