Summary

📚 What We've Learned in This Chapter

1. Introduction to Trigonometry

• Trigonometry is the study of relationships between the sides and angles of triangles
• The word "trigonometry" comes from Greek: "tri" (three), "gon" (sides), and "metron" (measure)
• Trigonometry has ancient origins and has been used by civilizations for thousands of years
• It has numerous real-world applications, from measuring heights of buildings to navigation systems

2. Trigonometric Ratios

In a right triangle ABC, right-angled at B:

• Sine of angle A (sin A) = Opposite side / Hypotenuse = BC/AC
• Cosine of angle A (cos A) = Adjacent side / Hypotenuse = AB/AC
• Tangent of angle A (tan A) = Opposite side / Adjacent side = BC/AB
• Cosecant of angle A (cosec A) = 1/sin A = Hypotenuse / Opposite side = AC/BC
• Secant of angle A (sec A) = 1/cos A = Hypotenuse / Adjacent side = AC/AB
• Cotangent of angle A (cot A) = 1/tan A = Adjacent side / Opposite side = AB/BC

The values of these ratios depend only on the angle, not on the size of the triangle.

3. Trigonometric Ratios of Specific Angles

We learned the exact values of trigonometric ratios for specific angles:

Angle	sin	cos	tan	cosec	sec	cot
0°	0	1	0	Not defined	1	Not defined
30°	1/2	√3/2	1/√3	2	2/√3	√3
45°	1/√2	1/√2	1	√2	√2	1
60°	√3/2	1/2	√3	2/√3	2	1/√3
90°	1	0	Not defined	1	Not defined	0

4. Trigonometric Identities

Important trigonometric identities include:

• Pythagorean Identities:

  sin² A + cos² A = 1
  tan² A + 1 = sec² A
  1 + cot² A = cosec² A

• Quotient Identities:

  tan A = sin A / cos A
  cot A = cos A / sin A

• Reciprocal Identities:

  cosec A = 1 / sin A
  sec A = 1 / cos A
  cot A = 1 / tan A

These identities are true for all angles (where the expressions are defined) and help us simplify complex trigonometric expressions.

💡 Key Points to Remember

1. Value Limitations:

   • The values of sin A and cos A are always between -1 and 1 (inclusive)
   • For acute angles, sin A and cos A are always positive
   • For acute angles, sec A and cosec A are always greater than or equal to 1

2. Complementary Angles:

   • If angles A and B are complementary (A + B = 90°), then:
     • sin A = cos B
     • cos A = sin B
     • tan A = cot B

3. Important Notes:

   • sin² A is short for (sin A)², not sin(sin A)
   • The trigonometric ratios of an angle depend only on the angle, not on the size of the triangle
   • As an angle increases from 0° to 90°, its sine increases from 0 to 1, and its cosine decreases from 1 to 0

🌍 Real-Life Applications

Trigonometry is used in various fields:

1. Architecture and Construction:

   • Calculating heights and distances of structures
   • Designing roofs, bridges, and other angular structures

2. Navigation:

   • Finding directions and locations using angles
   • GPS systems use trigonometric principles

3. Physics:

   • Analyzing waves, oscillations, and circular motion
   • Understanding force components in different directions

4. Astronomy:

   • Calculating distances to stars and planets
   • Determining the positions of celestial bodies

5. Computer Graphics:

   • Creating realistic animations and movements
   • Calculating rotations and projections in video games

🤔 Think About It!

Now that you've completed this chapter, consider these deeper questions:

1. How might the ancient Egyptians have used basic trigonometry to build the pyramids?
2. Why do you think the exact trigonometric values for most angles (like 10° or 22°) cannot be expressed as simple fractions?
3. How would our world be different if we didn't have trigonometry?
4. Can you think of any other real-life situations where you might use trigonometry?

🧠 Memory Aids

To help remember the concepts in this chapter:

1. SOHCAHTOA: A mnemonic for remembering the basic trigonometric ratios:

   • Sine = Opposite / Hypotenuse
   • Cosine = Adjacent / Hypotenuse
   • Tangent = Opposite / Adjacent

2. "All Students Take Calculus": For remembering which functions are positive in which quadrants (although we've focused on the first quadrant in this chapter)

3. Pattern in Values: Notice the pattern in sin and cos values for 0°, 30°, 45°, 60°, and 90°:

   • sin increases: 0, 1/2, 1/√2, √3/2, 1
   • cos decreases: 1, √3/2, 1/√2, 1/2, 0

🔜 What's Next?

In future mathematics courses, you'll explore:

• Trigonometric functions for angles greater than 90°
• Graphs of trigonometric functions
• Inverse trigonometric functions
• More complex trigonometric identities
• Applications of trigonometry in calculus and beyond

Congratulations on completing this introduction to trigonometry! The concepts you've learned here will serve as a foundation for many areas of mathematics and science.