Lecture Notes: Fundamental Trigonometric Identities

In this lecture, we discussed the fundamental trigonometric identities and applied them to solve a problem.

Definition

• Identity: An equation that is true for all values of the variable.
• In trigonometry, these equations remain valid for any angle θ where the function is defined.

Fundamental Trigonometric Identities

Reciprocal Identities

1. Cosecant (csc) Identity
   • cscθ = 1/sinθ
2. Secant (sec) Identity
   • secθ = 1/cosθ
3. Cotangent (cot) Identity
   • cotθ = 1/tanθ

Quotient Identities

1. Tangent (tan) Identity
   • tanθ = sinθ\cosθ
2. Cotangent (cot) Identity
   • cotθ = cosθ\sinθ

Example Problem

Given:

• sin(θ) = sqrt10 / 10
• cos(θ) = 3sqrt{10 / 10

Find the Four Remaining Trigonometric Functions

1. Cosecant (csc θ)

   • (\csc(θ) = \frac{1}{\sin(θ)} = \frac{1}{\left(\frac{\sqrt{10}}{10}\right)} = \frac{10}{\sqrt{10}})
   • Rationalize: (\frac{10}{\sqrt{10}} \times \frac{\sqrt{10}}{\sqrt{10}} = \sqrt{10})

2. Secant (sec θ)

   • (\sec(θ) = \frac{1}{\cos(θ)} = \frac{1}{\left(\frac{3\sqrt{10}}{10}\right)} = \frac{10}{3\sqrt{10}})
   • Rationalize: (\frac{10}{3\sqrt{10}} \times \frac{\sqrt{10}}{\sqrt{10}} = \frac{\sqrt{10}}{3})

3. Tangent (tan θ)

   • (\tan(θ) = \frac{\sin(θ)}{\cos(θ)} = \frac{\frac{\sqrt{10}}{10}}{\frac{3\sqrt{10}}{10}})
   • Simplify: (\frac{\sqrt{10}}{10} \times \frac{10}{3\sqrt{10}} = \frac{1}{3})

4. Cotangent (cot θ)

   • Using Reciprocal Identity: (\cot(θ) = \frac{1}{\tan(θ)} = \frac{1}{\frac{1}{3}} = 3)

Conclusion

• We successfully found the values of the remaining trigonometric functions using the fundamental identities.

Good Luck!