TR-13: The Trigonometric Ratios (Trigonometry series by Dennis F. Davis)

Introduction

This tutorial will guide you through the foundational concepts of trigonometric ratios as presented in the video by Dennis F. Davis. Understanding these ratios is essential for mastering trigonometry, especially in contexts such as geometry and physics. We will cover the construction of a standard-position acute angle triangle, the naming of sides, the mnemonic Soh-Cah-Toa, and the reciprocal relationships among the trigonometric functions.

Step 1: Construct the Standard-Position Acute Angle Triangle

1. Draw the Triangle: Start by sketching a right triangle on a coordinate plane.

   • Place the vertex of the right angle at the origin (0,0).
   • The angle of interest (acute angle) should be positioned in the first quadrant.

2. Label the Sides: Identify and label the sides of the triangle:

   • Opposite (Opp): The side opposite the acute angle.
   • Adjacent (Adj): The side next to the acute angle that is not the hypotenuse.
   • Hypotenuse (Hyp): The longest side, opposite the right angle.

Step 2: Understand Trigonometric Ratios

1. Define the Ratios: The six trigonometric ratios are based on the relationships between the sides of the triangle:

   • Sine (sin): Opp/Hyp
   • Cosine (cos): Adj/Hyp
   • Tangent (tan): Opp/Adj
   • Cosecant (csc): Hyp/Opp (reciprocal of sine)
   • Secant (sec): Hyp/Adj (reciprocal of cosine)
   • Cotangent (cot): Adj/Opp (reciprocal of tangent)

2. Use Soh-Cah-Toa: Remember the mnemonic to help recall the definitions:

   • Soh: Sine = Opposite / Hypotenuse
   • Cah: Cosine = Adjacent / Hypotenuse
   • Toa: Tangent = Opposite / Adjacent

Step 3: Identify Reciprocal Relationships

1. Reciprocal Functions: For each primary trigonometric function, the reciprocal function can be identified easily:

   • If you know sin, you can find csc by taking the reciprocal, i.e., csc = 1/sin.
   • Similarly, use this method for secant and cosecant, as well as tangent and cotangent.

2. Bridge Method: Utilize a simple bridge method to visualize and remember these relationships:

   • Draw arrows connecting each function to its reciprocal to create a visual map of relationships.

Step 4: Practice with Examples

1. Example Problems: Work on example problems to solidify your understanding. Here’s a basic exercise:

   • Given an angle with Opp = 3 and Hyp = 5, calculate:
     • sin = 3/5
     • cos = √(Hyp² - Opp²) / Hyp = √(5² - 3²) / 5 = √16 / 5 = 4/5
     • tan = Opp/Adj = 3/4 (after calculating Adj).

2. Drills: Utilize additional problems provided in the supplementary video TR-13X to practice various configurations of triangles and their corresponding trigonometric ratios.

Conclusion

Understanding trigonometric ratios is crucial for further studies in mathematics and science. By constructing a standard-position acute triangle and mastering the relationships between the sides, you lay the groundwork for more complex trigonometric concepts. For further practice, consider exploring the additional resources linked in the video description. Keep practicing to strengthen your skills!