How to solve an AAS triangle?
Understand the Problem
The question is asking how to solve a triangle given two angles and a side (AAS), which is a typical problem in trigonometry. To solve this type of triangle, we will typically use the known angles and side to find the remaining side lengths and angles using the laws of sines and cosines.
Answer
The sides and remaining angle can be calculated as follows: Angle $C = 180^\circ - A - B$ Side $b = a \cdot \frac{\sin B}{\sin A}$ Side $c = a \cdot \frac{\sin C}{\sin A}$
Answer for screen readers
The lengths of the sides and the measurement of angle $C$ are as follows:
- Length of side $b = a \cdot \frac{\sin B}{\sin A}$
- Length of side $c = a \cdot \frac{\sin C}{\sin A}$
- Angle $C = 180^\circ - A - B$
Steps to Solve
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Identify the Given Values You have two angles and one side. Let's denote the known angles as $A$ and $B$, and the known side as $a$. The missing angle $C$ can be calculated using the fact that the angles in a triangle sum to $180^\circ$: $$ C = 180^\circ - A - B $$
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Use the Law of Sines According to the Law of Sines, the ratios of the lengths of the sides of the triangle to the sines of the opposite angles are equal. This means we can find the lengths of the other two sides ($b$ and $c$) using the following formulas: $$ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} $$
To find side $b$, rearrange the equation: $$ b = a \cdot \frac{\sin B}{\sin A} $$
To find side $c$, use: $$ c = a \cdot \frac{\sin C}{\sin A} $$
- Calculate the Unknown Angles and Sides Using the known values, perform the calculations for angles and sides. Substitute the known values into the equations derived using the Law of Sines to find $b$ and $c$.
The lengths of the sides and the measurement of angle $C$ are as follows:
- Length of side $b = a \cdot \frac{\sin B}{\sin A}$
- Length of side $c = a \cdot \frac{\sin C}{\sin A}$
- Angle $C = 180^\circ - A - B$
More Information
In triangles where two angles and a side are known (AAS), the solution relies heavily on the Law of Sines. It is important to remember that knowing two angles allows you to find the third angle. Trigonometry is widely used in architecture, engineering, and many fields of science.
Tips
- Forgetting to convert angles to the correct format (degrees or radians) when calculating trigonometric functions.
- Incorrectly summing the angles leading to an incorrect measurement of angle $C$.
- Misapplying the Law of Sines, especially with the order of substitutions.
