Problem 22 - Algebra and Trigonometry

Question

Solve each triangle. $$ b=4, \quad c=1, \quad A=120^{\circ} $$

Step-by-Step Solution

Answer

a = \sqrt{21}, B \approx 15^{\circ}, C \approx 45^{\circ}

Step 1: Apply the Law of Cosines

Use the Law of Cosines to find side a. The Law of Cosines states: \[a = \sqrt{b^{2} + c^{2} - 2bc \cos(A)}\] Plug in the given values: \[a = \sqrt{4^{2} + 1^{2} - 2 \cdot 4 \cdot 1 \cos(120^{\circ})}\]

Step 2: Calculate the Cosine Value

Finding \cos(120^{\circ}) which is -1/2. Substitute this into the equation: \[a = \sqrt{4^{2} + 1^{2} - 2 \cdot 4 \cdot 1 \cdot \left(-\frac{1}{2}\right)}\]

Step 3: Simplify the Expression

Simplify the expression inside the square root: \[a = \sqrt{16 + 1 + 4} = \sqrt{21} \approx 4.58\]

Step 4: Apply the Law of Sines

Use the Law of Sines to find the other angles. The Law of Sines states: \[\frac{\sin(B)}{b} = \frac{\sin(A)}{a}\] or \[\frac{\sin(B)}{4} = \frac{\sin(120^{\circ})}{\sqrt{21}}\]

Step 5: Solve for Angle B

Find \sin(120^{\circ}) which is \sqrt{3}/2. Therefore, \[\sin(B) = \frac{4 \sqrt{3}}{2\sqrt{21}}\]

Step 6: Calculate Angle B and C

Solve for B using \sin^{-1}(value). Then, subtract angles A and B from 180 to find C. \[C = 180^{\circ} - 120^{\circ} - B\]

Key Concepts

solving triangles

Understanding how to solve triangles is crucial in trigonometry. It involves determining the unknown side lengths and angles of a triangle using given information. Let me break this down for you:
1. We start with the information provided about the triangle, such as side lengths and angles.
2. Depending on the given data, we choose the appropriate trigonometric formulas to find the unknowns.
In the problem we have here, we know two sides, **b** and **c**, and one angle, **A**. To solve for the unknown side and the other angles, follow these steps:

• Use the Law of Cosines to find the unknown side.
• Once the unknown side is found, use the Law of Sines to find the other two angles.

trigonometric identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the included variables. The key identities used in triangle problems include:

• The sine function, \(\text{sin}(\theta)\).
• The cosine function, \(\text{cos}(\theta)\).
• The tangent function, \(\text{tan}(\theta)\).

For our problem, we specifically use the cosine function when applying the Law of Cosines and the sine function with the Law of Sines. For example, \(\text{cos}(120^{\circ}) = -\frac{1}{2}\) and \(\text{sin}(120^{\circ}) = \frac{\sqrt{3}}{2}\). These values were crucial for solving the triangle.

Law of Sines

The Law of Sines is very useful for solving triangles, especially when dealing with non-right triangles. It states that the ratio of the length of a side to the sine of the opposite angle is constant for all three sides and angles in the triangle:
every triangle
\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]
In our case, after finding side **a** using the Law of Cosines, we use the Law of Sines to find angles \(B\) and \(C\). Specifically, we use:
\[\frac{\sin B}{b} = \frac{\sin(120^{\circ})}{a}\]
Plugging in the values we have, enables us to solve for \(\sin B\) and then use the inverse sine function \((\sin^{-1})\) to find the measure of angle \(B\).

angle calculation

Finally, calculating the angles is a critical step. To find angle **B**, we use the calculated value from the Law of Sines:
\[\sin B = \frac{4 \sqrt{3}}{2\sqrt{21}}\]
Then apply the arc sine (or inverse sine) function:
\[B = \sin^{-1}{\left( \frac{4 \sqrt{3}}{2\sqrt{21}} \right)}\]
Once you have angle **B**, finding angle **C** is straightforward:
\[C = 180^{\circ} - 120^{\circ} - B\]
This makes sure that the sum of all angles in the triangle equals \(180^{\circ}\), as required for any triangle.