Trigonometric Identities Overview

Introduction

• Common trig identities useful for trigonometry courses.
• Applicable for both beginners and exam preparation.

Key Trigonometric Ratios (SOHCAHTOA)

1. Sine (sin):
   [ \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} ]
2. Cosine (cos):
   [ \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} ]
3. Tangent (tan):
   [ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} ]

Reciprocal Identities

• Cosecant (csc):
  [ \csc(\theta) = \frac{1}{\sin(\theta)} ]
• Secant (sec):
  [ \sec(\theta) = \frac{1}{\cos(\theta)} ]
• Cotangent (cot):
  [ \cot(\theta) = \frac{1}{\tan(\theta)} ]

Example Problem: 3-4-5 Triangle

• Sine:
  [ \sin(\theta) = \frac{4}{5} ]
• Cosine:
  [ \cos(\theta) = \frac{3}{5} ]
• Tangent:
  [ \tan(\theta) = \frac{4}{3} ]

Quotient Identities

• [ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} ]
• [ \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} ]

Pythagorean Identities

1. [ \sin^2(\theta) + \cos^2(\theta) = 1 ]
2. [ 1 + \cot^2(\theta) = \csc^2(\theta) ]
3. [ 1 + \tan^2(\theta) = \sec^2(\theta) ]

Even and Odd Functions

• Even Functions:
  • Cosine (cos)
  • Secant (sec)
• Odd Functions:
  • Sine (sin)
  • Tangent (tan)
  • Cosecant (csc)
  • Cotangent (cot)

Cofunction Identities

• [ \cos(\theta) = \sin(90^\circ - \theta) ]
• [ \sin(\theta) = \cos(90^\circ - \theta) ]
• Other cofunction identities include secant-cosecant and tangent-cotangent pairs.

Double Angle Identities

1. [ \sin(2\theta) = 2\sin(\theta)\cos(\theta) ]
2. [ \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) ]
3. [ \tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)} ]

Half Angle Identities

1. [ \sin\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos(\theta)}{2}} ]
2. [ \cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos(\theta)}{2}} ]
3. [ \tan\left(\frac{\theta}{2}\right) = \frac{\sqrt{1 - \cos(\theta)}}{1 + \cos(\theta)} ]

Sum and Difference Identities

1. [ \sin(\alpha \pm \beta) = \sin(\alpha)\cos(\beta) \pm \cos(\alpha)\sin(\beta) ]
2. [ \cos(\alpha \pm \beta) = \cos(\alpha)\cos(\beta) \mp \sin(\alpha)\sin(\beta) ]
3. [ \tan(\alpha \pm \beta) = \frac{\tan(\alpha) \pm \tan(\beta)}{1 \mp \tan(\alpha)\tan(\beta)} ]

Power Reducing Formulas

1. [ \sin^2(\theta) = \frac{1 - \cos(2\theta)}{2} ]
2. [ \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} ]
3. [ \tan^2(\theta) = \frac{1 - \cos(2\theta)}{1 + \cos(2\theta)} ]

Product to Sum Formulas

1. [ \sin(\alpha)\sin(\beta) = \frac{1}{2}[\cos(\alpha - \beta) - \cos(\alpha + \beta)] ]
2. [ \cos(\alpha)\cos(\beta) = \frac{1}{2}[\cos(\alpha - \beta) + \cos(\alpha + \beta)] ]
3. [ \sin(\alpha)\cos(\beta) = \frac{1}{2}[\sin(\alpha + \beta) + \sin(\alpha - \beta)] ]

Law of Sines

• [ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} ]

Law of Cosines

• [ c^2 = a^2 + b^2 - 2ab\cos(C) ]

Area of a Triangle

• Area = [ \frac{1}{2}ab\sin(C) ]
• Heron's Formula:
  • [ s = \frac{a + b + c}{2} ]
  • Area = [ \sqrt{s(s - a)(s - b)(s - c)} ]

Conclusion

• Overview of essential trigonometric identities and formulas for course and exam preparation.