Introduction to the trigonometric ratios

Here are the key points to learn when studying "Triangle Similarity & Trigonometric Ratios":

Triangle Similarity

1. Definition: Two triangles are similar if their corresponding angles are equal and their corresponding sides are in proportion.

2. Criteria for Similarity:

   • AA (Angle-Angle) Criterion: If two angles of one triangle are equal to two angles of another triangle, the triangles are similar.
   • SSS (Side-Side-Side) Criterion: If the sides of two triangles are in proportion, the triangles are similar.
   • SAS (Side-Angle-Side) Criterion: If two sides of one triangle are proportional to two sides of another triangle and the included angles are equal, the triangles are similar.

3. Properties of Similar Triangles:

   • Corresponding sides are in the same ratio.
   • The ratio of areas is the square of the ratio of corresponding sides.

Trigonometric Ratios

1. Basic Ratios: For a right triangle, the primary trigonometric ratios are:

   • Sine (sin): Ratio of the opposite side to the hypotenuse (sin = opposite/hypotenuse).
   • Cosine (cos): Ratio of the adjacent side to the hypotenuse (cos = adjacent/hypotenuse).
   • Tangent (tan): Ratio of the opposite side to the adjacent side (tan = opposite/adjacent).

2. Reciprocal Ratios:

   • Cosecant (csc): Reciprocal of sine (csc = 1/sin = hypotenuse/opposite).
   • Secant (sec): Reciprocal of cosine (sec = 1/cos = hypotenuse/adjacent).
   • Cotangent (cot): Reciprocal of tangent (cot = 1/tan = adjacent/opposite).

3. Pythagorean Identity:

   • The relationship $\sin^2(θ) + \cos^2(θ) = 1$ is foundational for understanding and deriving other trigonometric concepts.

4. Applications:

   • Trigonometric ratios are used to find unknown side lengths or angles in right triangles.
   • They are instrumental in solving problems involving heights, distances, and angles in various fields such as physics, engineering, and architecture.

Conclusion

Understanding triangle similarity allows for the application of proportional reasoning and the properties of geometric figures, while mastering trigonometric ratios is essential for solving practical problems involving angles and lengths in right triangles.

Sure! Here are the key points to learn when studying trigonometric ratios in right triangles:

1. Definition of Trigonometric Ratios:

   • Sine (sin): Ratio of the opposite side to the hypotenuse (sin(θ) = opposite/hypotenuse).
   • Cosine (cos): Ratio of the adjacent side to the hypotenuse (cos(θ) = adjacent/hypotenuse).
   • Tangent (tan): Ratio of the opposite side to the adjacent side (tan(θ) = opposite/adjacent).

2. Basic Right Triangle Properties:

   • A right triangle has one angle that is 90 degrees.
   • The sides of the triangle are referred to as the opposite (relative to the angle), adjacent, and hypotenuse (the longest side).

3. Use of the Ratios:

   • These ratios can be used to find unknown side lengths or angles in right triangles.
   • They are foundational for solving problems in geometry and trigonometry.

4. Reciprocal Ratios:

   • Cosecant (csc): Reciprocal of sine (csc(θ) = 1/sin(θ)).
   • Secant (sec): Reciprocal of cosine (sec(θ) = 1/cos(θ)).
   • Cotangent (cot): Reciprocal of tangent (cot(θ) = 1/tan(θ)).

5. Applications:

   • Trigonometric ratios are used in various fields such as physics, engineering, and architecture.
   • They are essential for solving real-world problems involving heights, distances, and angles.

6. Unit Circle Connection:

   • Understanding how trigonometric ratios relate to the unit circle for angles beyond 90 degrees.

7. Angles of Elevation and Depression:

   • Applications of trigonometric ratios to solve problems involving angles of elevation and depression.

8. Calculator Use:

   • Familiarity with using calculators to compute trigonometric functions for angle measures.

By mastering these concepts, you will have a solid understanding of trigonometric ratios in right triangles!