91Ó°ÊÓ

Solving a triangle involves finding all its angles and sides. When you're given certain angles and one side, like side \( a = 420 \) and angles \( A = 22^{\circ} \) and \( B = 95^{\circ} \), it's about figuring out the rest using known formulas. Here's a step-by-step guide on how to do it:

• Identify given information: Start by noting down the given angles and side.
• Find the missing angle: Use the Angle Sum Property (more on this later) to find the third angle.
• Apply the Law of Sines: This will help you find the other two sides.
• Verify and sketch: Finally, sketch the triangle to ensure it makes sense visually.

The Angle Sum Property is a fundamental rule in geometry applied to triangles. It states that the sum of the interior angles in a triangle is always equal to \( 180^{\circ} \).For instance, if you know two angles of a triangle, you can easily determine the third:- Given \( \angle A = 22^{\circ} \) and \( \angle B = 95^{\circ} \), you can find the third angle using: \[ \angle C = 180^{\circ} - \angle A - \angle B = 180^{\circ} - 22^{\circ} - 95^{\circ} = 63^{\circ}. \]This property is essential for solving triangles because it provides a straightforward method to find an unknown angle, which is crucial for subsequent calculations.

Sketching triangles from given data helps visualize the problem and verify your calculations. Start with a rough sketch based on the angles and side you know. Here’s how:- Draw the known side: In our problem, since \( a = 420 \) is given, draw this line first.- Mark given angles: Angle \( A = 22^{\circ} \) and \( \angle B = 95^{\circ} \) are placed correctly following the known sides.- Complete the triangle: Use the Angle Sum Property to place the final angle \( \angle C = 63^{\circ} \).Ensure the sketch represents the angles correctly—this visual confirmation can often highlight any discrepancies in your results before you go further.

Trigonometry is a branch of mathematics that studies the relationships between the angles and sides of triangles. In this problem, the Law of Sines is particularly useful:This law is expressed as:\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}, \]where \( a \), \( b \), and \( c \) are sides opposite angles \( A \), \( B \), and \( C \) respectively. It's used to find unknown sides or angles when some of the triangle's characteristics are known. For example, to find side \( b \) we rearrange the formula:\[ b = \frac{420 \cdot \sin 95^{\circ}}{\sin 22^{\circ}} \approx 1116.96. \]This formula can be used similarly to find \( c \) and is crucial in solving any non-right triangle accurately.