Trigonometric Identities Lecture Notes

Overview

• Common trig identities discussed
• Useful for beginners and final exam preparation

Basic Trig Ratios (SOHCAHTOA)

• Sine: ( ext{sine} \Theta = \frac{\text{opposite}}{\text{hypotenuse}} )
• Cosine: ( ext{cosine} \Theta = \frac{\text{adjacent}}{\text{hypotenuse}} )
• Tangent: ( ext{tangent} \Theta = \frac{\text{opposite}}{\text{adjacent}} )

Reciprocal Identities

• Cosecant: ( \text{cosecant} \Theta = \frac{1}{\text{sine} \Theta} )
• Secant: ( \text{secant} \Theta = \frac{1}{\text{cosine} \Theta} )
• Cotangent: ( \text{cotangent} \Theta = \frac{1}{\text{tangent} \Theta} )

Example Problem

For a 3-4-5 Triangle

• Sine: ( \text{sine} \Theta = \frac{4}{5} )
• Cosine: ( \text{cosine} \Theta = \frac{3}{5} )
• Tangent: ( \text{tangent} \Theta = \frac{4}{3} )
• Cosecant: ( \text{cosecant} \Theta = \frac{5}{4} )
• Secant: ( \text{secant} \Theta = \frac{5}{3} )
• Cotangent: ( \text{cotangent} \Theta = \frac{3}{4} )

Quotient Identities

• ( ext{tangent} \Theta = \frac{\text{sine} \Theta}{\text{cosine} \Theta} )
• ( ext{cotangent} \Theta = \frac{\text{cosine} \Theta}{\text{sine} \Theta} )

Pythagorean Identities

1. ( \text{sine}^2 \Theta + \text{cosine}^2 \Theta = 1 )
2. ( 1 + \text{cotangent}^2 \Theta = \text{cosecant}^2 \Theta )
3. ( 1 + \text{tangent}^2 \Theta = \text{secant}^2 \Theta )

Even and Odd Functions

• Even Functions: ( ext{cosine} \Theta, \text{secant} \Theta )
• Odd Functions: ( ext{sine} \Theta, \text{tangent} \Theta, \text{cosecant} \Theta, \text{cotangent} \Theta )

Cofunction Identities

• ( \text{cosine} \Theta = \text{sine}(90^\circ - \Theta) )
• ( ext{cosecant} \Theta = \text{secant}(90^\circ - \Theta) )
• ( ext{cotangent} \Theta = \text{tangent}(90^\circ - \Theta) )

Double Angle Identities

• ( ext{sine}(2\Theta) = 2 \cdot \text{sine} \Theta \cdot \text{cosine} \Theta )
• ( ext{cosine}(2\Theta) = \text{cosine}^2 \Theta - \text{sine}^2 \Theta ) (also ( 2 \cdot \text{cosine}^2 \Theta - 1 ) and ( 1 - 2 \cdot \text{sine}^2 \Theta ))
• ( ext{tangent}(2\Theta) = \frac{2 \cdot \text{tangent} \Theta}{1 - \text{tangent}^2 \Theta} )

Half Angle Identities

• ( \text{sine}(\frac{\Theta}{2}) = \pm \sqrt{\frac{1 - \text{cosine} \Theta}{2}} )
• ( \text{cosine}(\frac{\Theta}{2}) = \pm \sqrt{\frac{1 + \text{cosine} \Theta}{2}} )
• ( \text{tangent}(\frac{\Theta}{2}) = \frac{\sqrt{1 - \text{cosine} \Theta}}{1 + \text{cosine} \Theta} )

Sum and Difference Identities

• ( \text{sine}(\alpha \pm \beta) = \text{sine} \alpha \cdot \text{cosine} \beta \pm \text{cosine} \alpha \cdot \text{sine} \beta )
• ( \text{cosine}(\alpha \pm \beta) = \text{cosine} \alpha \cdot \text{cosine} \beta \mp \text{sine} \alpha \cdot \text{sine} \beta )
• ( \text{tangent}(\alpha \pm \beta) = \frac{\text{tangent} \alpha \pm \text{tangent} \beta}{1 \mp \text{tangent} \alpha \cdot \text{tangent} \beta} )

Power Reducing Formulas

• ( ext{sine}^2 \Theta = \frac{1 - \text{cosine}(2\Theta)}{2} )
• ( ext{cosine}^2 \Theta = \frac{1 + \text{cosine}(2\Theta)}{2} )
• ( ext{tangent}^2 \Theta = \frac{1 - \text{cosine}(2\Theta)}{1 + \text{cosine}(2\Theta)} )

Product to Sum Formulas

• ( ext{sine} \alpha \cdot ext{sine} \beta = \frac{1}{2} [\text{cosine}(\alpha - \beta) - \text{cosine}(\alpha + \beta)] )
• ( ext{cosine} \alpha \cdot ext{cosine} \beta = \frac{1}{2} [\text{cosine}(\alpha - \beta) + \text{cosine}(\alpha + \beta)] )
• ( ext{sine} \alpha \cdot ext{cosine} \beta = \frac{1}{2} [\text{sine}(\alpha + \beta) + \text{sine}(\alpha - \beta)] )

Law of Sines

• For triangle: ( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} )

Law of Cosines

• ( c^2 = a^2 + b^2 - 2ab \cdot \text{cos}(C) )

Area of Triangle

• Area = ( \frac{1}{2} ab \cdot ext{sine}(C) )
• Heron's Formula: Area = ( \sqrt{s(s-a)(s-b)(s-c)} ) where ( s = \frac{a+b+c}{2} )

Law of Tangents

• ( \frac{a - b}{a + b} = \tan\left(\frac{A - B}{2}\right) ) or ( \frac{a - b}{a + b} = \tan\left(\frac{A + B}{2}\right) )

Conclusion

• Trigonometric identities essential for study and practical application in courses and exams.