(97) Trig Identities

Chapter 1: Introduction to Trigonometric Identities

• Common trig identities useful for trigonometry courses and exams.

Right Triangle Basics

• Key expression: SOHCAHTOA

  • SOH: Sine (Sin(Θ) = Opposite/Hypotenuse)

  • CAH: Cosine (Cos(Θ) = Adjacent/Hypotenuse)

  • TOA: Tangent (Tan(Θ) = Opposite/Adjacent)

Reciprocal Identities

• Cosecant (csc): Csc(Θ) = 1/Sin(Θ)

• Secant (sec): Sec(Θ) = 1/Cos(Θ)

• Cotangent (cot): Cot(Θ) = 1/Tan(Θ)

Example Problem: 3-4-5 Triangle

• Sin(Θ) = 4/5

• Cos(Θ) = 3/5

• Tan(Θ) = 4/3

• Csc(Θ) = 5/4

• Sec(Θ) = 5/3

• Cot(Θ) = 3/4

Quotient Identities

• Tan(Θ) = Sin(Θ)/Cos(Θ)

• Cot(Θ) = Cos(Θ)/Sin(Θ)

Pythagorean Identities

1. Sin²(Θ) + Cos²(Θ) = 1

2. 1 + Cot²(Θ) = Csc²(Θ)

3. 1 + Tan²(Θ) = Sec²(Θ)

Chapter 2: Even and Odd Functions

• Odd Functions:

  • Sin(−Θ) = −Sin(Θ)

  • Tan(−Θ) = −Tan(Θ)

  • Csc(−Θ) = −Csc(Θ)

  • Cot(−Θ) = −Cot(Θ)

• Even Functions:

  • Cos(−Θ) = Cos(Θ)

  • Sec(−Θ) = Sec(Θ)

Chapter 3: Co-Function Identities

• Cos(Θ) = Sin(90° − Θ)

• Sin(Θ) = Cos(90° − Θ)

• Csc(Θ) = Sec(90° − Θ)

• Cot(Θ) = Tan(90° − Θ)

Double Angle Identities

• Sin(2Θ) = 2Sin(Θ)Cos(Θ)

• Cos(2Θ) = Cos²(Θ) − Sin²(Θ)

• Tan(2Θ) = (2Tan(Θ))/(1 − Tan²(Θ))

Chapter 4: Half Angle Formulas

• Sin(Θ/2) = ±√((1 − Cos(Θ))/2)

• Cos(Θ/2) = ±√((1 + Cos(Θ))/2)

• Tan(Θ/2) = ±√((1 − Cos(Θ))/(1 + Cos(Θ)))

  • Alternatives:

  • Tan(Θ/2) = (1 − Cos(Θ))/Sin(Θ)

  • Tan(Θ/2) = Sin(Θ)/(1 + Cos(Θ))

Sum and Difference Identities

• Sin(A ± B) = Sin(A)Cos(B) ± Cos(A)Sin(B)

• Cos(A ± B) = Cos(A)Cos(B) ∓ Sin(A)Sin(B)

• Tan(A ± B) = (Tan(A) ± Tan(B))/(1 ∓ Tan(A)Tan(B)

Chapter 5: Power Reducing Formulas

• Sin²(Θ) = (1 − Cos(2Θ))/2

• Cos²(Θ) = (1 + Cos(2Θ))/2

• Tan²(Θ) = (1 − Cos(2Θ))/(1 + Cos(2Θ))

Product to Sum Formulas

• Sin(A)Sin(B) = 1/2 [Cos(A − B) − Cos(A + B)]

• Cos(A)Cos(B) = 1/2 [Cos(A − B) + Cos(A + B)]

• Sin(A)Cos(B) = 1/2 [Sin(A + B) + Sin(A − B)]

Sum to Product Formulas

• Sin(A + Sin(B)) = 2Sin((A + B)/2)Cos((A − B)/2)

Chapter 6: Law of Sines and Cosines

• Law of Sines:

  • Sin(A)/a = Sin(B)/b = Sin(C)/c

• Law of Cosines:

  • c² = a² + b² − 2abCos(C)

  • Use to find angles when all sides are known.

Area of Triangle

• Area = 1/2abSin(C)

• Heron’s formula: Area = √[s(s − a)(s − b)(s − c)], where s = (a + b + c)/2

Law of Tangents

• (a − b)/(a + b) = Tan[(1/2)(α − β)]

Conclusion

• Summary of important trigonometric formulas and identities for study.