Introduction to Sine Law and Trigonometry

Study Notes

Introduction to Sine Law

Trigonometry for right triangles uses trigonometric ratios (SOH CAH TOA) and only applies to right-angled triangles.
For triangles without a right angle, alternative methods are needed, like splitting into right triangles or using the Sine Law.
Acute angled triangles have all angles less than 90 degrees.

Using Right Triangles to Determine Sides

Acute angled triangles can be split into two right triangles to find side lengths.
Use trigonometric ratios (SOH CAH TOA) based on known sides and the reference angle.
Identify opposite, adjacent, and hypotenuse sides relative to the reference angle.

Intermediate Steps in Right Triangles

Sometimes, finding a side length requires an intermediate step using another right triangle.
This involves finding a common side length connecting the two right triangles.
This shared side is often adjacent to one reference angle and a different reference angle in another triangle.

Sine Law

The Sine Law states that in any triangle, the ratio of a side to the sine of its opposite angle is constant.
This means for any three sides and their opposite angles: a/sin A = b/sin B = c/sin C.
The Sine Law can also be expressed as sin A/a = sin B/b = sin C/c.

Proving the Sine Law

The Sine Law is proven by considering two right triangles formed within a triangle.
The sine of an angle in each right triangle equals a common side length divided by the hypotenuse.
Manipulating these equations and equating the common side length derives the Sine Law.

Using the Sine Law for Side Lengths

To find a side length using the Sine Law, know:
- Two sides and their opposite angles.
- Two angles and a side opposite one of the angles.

Using the Sine Law for Angle Measures

To find an angle using the Sine Law, know:
- Two sides and the angle opposite one of the sides.
- Two angles and one side.

Applications of the Sine Law

Sine Law is used in surveying, navigation, and engineering.
It allows finding unknown side lengths and angle measures in any triangle.
This enables problem-solving in real-world situations involving triangles.

Example Problems

Example 1: Using two right triangles to determine a side length (e.g., BC).
Example 2: Using the Sine Law to calculate a side length (e.g., BC).
Example 3: Using the Sine Law to find an unknown angle, requiring an intermediate step.
Example 4: Applying the Sine Law to determine the perimeter of a triangle based on sides and angles.

Description

This quiz explores the concepts of the Sine Law and its applications in triangles without right angles, focusing on acute angled triangles. Understand how to utilize right triangles and trigonometric ratios (SOH CAH TOA) to determine side lengths effectively.