Problem 29 - Algebra and Trigonometry

Question

If two angles and the included side are given, the third angle is easy to find. Use the Law of sines to show that the area \(K\) of a triangle with side \(a\) and angles \(A, B,\) and \(C\) is $$K=\frac{a^{2} \sin B \sin C}{2 \sin A}$$

Step-by-Step Solution

Answer

The area of the triangle is given by the formula K=\frac{a^2\sin B \sin C}{2 \sin A}.

Step 1 - Recall the Law of Sines

The Law of Sines states that \[\begin{equation}\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}divide all the sides by sin(A), sin(B), sin(C), respectivelyand obtain the relation between the sides and angles.\end{equation}\].

Step 2 - The formula for the area of a triangle using sides and sine

The area of a triangle can be expressed as \[\begin{equation}K = \frac{1}{2} a b \sin CUsing the Law of Sines,\frac{a}{\sin A} = \frac{b}{\sin B}, replace side bb = \frac{a \sin B}{\sin A}Now substitute b\frac{1}{2} a \left (\frac{a \sin B}{\sin A} \right) \sin C\end{equation}\].

Step 3 - Simplify the Expression

Simplify the expression obtained from step 2\[\begin{equation}K = \frac{1}{2} \cdot \frac{a^2 \sin B \sin C}{\sin A}ewline \boxed{K=\frac{a^2\sin B \sin C}{2 \sin A}}\end{equation}\]

Key Concepts

triangle area

Understanding how to calculate the area of a triangle is essential in many fields. One useful formula to find the area \(K\) of a triangle when given two angles and the included side uses trigonometry. This formula is \[ K = \frac{1}{2} ab \sin C \] where \(a\) and \(b\) are two sides and \(C\) is the included angle. This step is derived from the basic trigonometric definition involving the sine function.
For a deeper insight, we use the Law of Sines to transform and simplify as shown later. This leads to a more versatile formula useful in various scenarios.

trigonometric identities

Trigonometric identities are essential tools in solving many geometry problems. They help to simplify complex expressions. The Law of Sines is one such identity:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \] This equation reveals the relationship between the sides and angles of a triangle, allowing us to transition from one set of known values to another.
Here, the identity helps replace side \(b\) with an expression involving side \(a\), simplifying our initial area formula. These identities are foundational in many proofs and simplify many trigonometric equations in the context of both triangle geometry and more advanced topics.

sine rule

The Sine Rule or Law of Sines is a powerful trigonometric formula linking the angles and sides of a triangle. In our task, it helps transform the area formula into a more flexible format.
Recall the steps to derive the area formula using the Law of Sines:

• Step 1: Recall the identity \[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]
• Step 2: Express the area \(K\) using side \(a\) and angle \(B\)
• Step 3: Substituting, we find \[ K = \frac{1}{2} \cdot \frac{a^2 \sin B \sin C}{\sin A} \].
  This derivation relies on the Law of Sines to replace \(b\), simplifying our area equation to the final form

The flexibility and simplicity of the Sine Rule make it an invaluable tool in trigonometry, particularly in solving triangles where standard formulas might be cumbersome.