Question

What do you do first if you are asked to solve a triangle and are given two sides and the included angle?

Short answer

First, apply the Law of Cosines to find the third side. Then use the Law of Sines to find one of the unknown angles. Finally, calculate the remaining angle.

Step-by-step solution

Identify the Given Elements

Determine which two sides and which included angle are provided in the problem. Label these parts on the diagram of the triangle if available.

Apply the Law of Cosines

Use the Law of Cosines to find the length of the third side. The Law of Cosines is given by \( c^2 = a^2 + b^2 - 2ab \cos(C) \), where \ a \ and \ b \ are the known sides, and \ C \ is the included angle.

Calculate the Third Side

Plug in the known values into the Law of Cosines formula and solve for the third side.

Use the Law of Sines

Now use the Law of Sines to find one of the unknown angles. The Law of Sines states that \[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \]. Use the known sides and included angle to solve for one of the other angles.

Calculate the Remaining Angle

Subtract the sum of the known angles from 180 degrees to find the third angle in the triangle.

Key concepts

Law of Cosines

The Law of Cosines is instrumental in solving triangles when you have two sides and the included angle (SAS case). It helps to find the length of the unknown side. The formula for the Law of Cosines is:
\[ c^2 = a^2 + b^2 - 2ab \, \text{cos}(C) \]
In this equation:

• a and b are the known sides.
• C is the included angle.
• c is the side opposite to the angle C, which we want to find.

By plugging the known values into the formula, you can calculate the third side of the triangle.

Law of Sines

The Law of Sines is incredibly helpful in finding unknown angles or sides, especially when you start with two sides and an included angle, after using the Law of Cosines. The Law of Sines is given by the relation:
\[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \]
In this equation:

• a, b, and c are the sides of the triangle.
• A, B, and C are the angles opposite these sides respectively.

After finding the third side using the Law of Cosines, use the Law of Sines to find an unknown angle. Substitute the known sides and their corresponding angles into the equation, solve for the missing angle.

Triangle Angles

Every triangle has three angles, and the sum of these angles is always 180 degrees. This is a fundamental property of triangles.

• Firstly, identify the given angle and the two sides.
• Secondly, use the Law of Cosines to find the unknown side.
• Thirdly, apply the Law of Sines to find an unknown angle.
• Finally, subtract the sum of the known angles from 180 degrees to find the remaining angle.

By following these steps, you ensure that all angles are correctly calculated, maintaining the triangle's properties.

Included Angle

The 'included angle' refers to the angle formed between two known sides of a triangle. For example, if sides a and b are known, the included angle is the angle between these sides, designated as C in the examples.
Understanding how to use the included angle is crucial when applying both the Law of Cosines and the Law of Sines. Knowledge of this angle allows you to:

• Determine the length of the unknown side using the Law of Cosines.
• Find unknown angles, especially when combined with known side lengths, using the Law of Sines.

It's a key starting point for solving triangles with given sides and an included angle.

Trigonometry

Trigonometry is the branch of mathematics that focuses on the relationships between the angles and sides of triangles. The fundamentals used here, like the Law of Cosines and the Law of Sines, are central concepts in trigonometry. Through trigonometry, we solve for:

• Unknown sides in a triangle when certain angles and sides are given.
• Unknown angles when certain sides and angles are known.

With principles such as trigonometric functions (sine, cosine), you can navigate through complex triangular problems confidently.
It's beneficial to get comfortable with these key relationships to ease solving triangle-related problems in your studies and beyond.