Trigonometry Class: Ratios and Functions

Basic Trigonometric Ratios

• Trigonometric ratios relate the angles of a right-angled triangle to the ratios of its side lengths. They are fundamental in trigonometry.
• The most common trigonometric ratios are sine (sin), cosine (cos), and tangent (tan).
• These ratios are defined for an acute angle θ in a right-angled triangle.
• Sine (sin θ): The ratio of the length of the side opposite to the angle θ to the length of the hypotenuse. sin θ = opposite / hypotenuse
• Cosine (cos θ): The ratio of the length of the side adjacent to the angle θ to the length of the hypotenuse. cos θ = adjacent / hypotenuse
• Tangent (tan θ): The ratio of the length of the side opposite to the angle θ to the length of the side adjacent to the angle θ. tan θ = opposite / adjacent

Reciprocal Trigonometric Ratios

• There are three reciprocal trigonometric ratios: cosecant (csc), secant (sec), and cotangent (cot).
• Cosecant (csc θ): The reciprocal of sine. csc θ = 1 / sin θ
• Secant (sec θ): The reciprocal of cosine. sec θ = 1 / cos θ
• Cotangent (cot θ): The reciprocal of tangent. cot θ = 1 / tan θ

Important Relationships

• The Pythagorean identity relates the sine and cosine functions: sin²θ + cos²θ = 1
• This identity holds true for all angles θ.
• Using the definitions of sine, cosine and tangent, and the Pythagorean Theorem, we can derive the following relationships:
• tan θ = sin θ / cos θ
• cot θ = cos θ / sin θ

Usage in Solving Right-Angled Triangles

• Trigonometric ratios are used to find unknown side lengths and angles in right-angled triangles.
• Given one side length and an acute angle, you can find other side lengths and angles using the appropriate ratios.
• The choice of which ratio to use depends on the known and unknown values.
• Example problems may involve finding the length of a missing side or the measure of a missing angle using a known side and a known angle, or two known sides.

Beyond Right-Angled Triangles

• Trigonometric ratios can be extended to obtuse angles and angles larger than 90 degrees.
• This involves incorporating the trigonometric definitions into specific quadrants in the coordinate plane, enabling the use of values for angles outside the 0° to 90° range.

Applications

• Trigonometric ratios are used in many fields, including:
  • Engineering: Designing structures, calculating distances, navigating.
  • Navigation: Determining bearings, plotting courses.
  • Surveying: Measuring land areas, determining heights.
  • Physics: Calculating forces, velocities, and other physical quantities.
• A practical application is determining the height of a building using the angle of elevation and the distance from the building to the observer.