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Theorem 3: Derivatives of Trigonometric Functions

THEOREM 3

(i) d(sinθ) = cos θ dθ,                  d(cos θ) =   -sin θ dθ,

(ii) d(tan θ) = sec2 θ dθ,                d(cot θ) =  -csc2 θ dθ,

(iii) d(sec θ) = sec θ tan θ dθ,        d(csc θ) =  -csc θ cot θ dθ.

PROOF

We prove the formula for d(tan θ) and leave the rest as problems.

07_trigonometric_functions-74.gif

These formulas lead at once to new integration formulas.

(i)

d(sinθ) = cos θ dθ,

d(cot θ) = - sin θ dθ,

(ii)

d(tan θ) = sec2 θ dθ,

d(cot θ) = -esc2 θdθ,

(iii)

d(sec θ) = sec θ tan θ dθ.

d(csc θ) = - esc θ cot θ dθ.

We are not yet able to evaluate the integrals ∫ tan θ dθ, ∫ cot θ dθ, ∫ sec θ dθ, ∫ csc θ dθ. These integrals will be found in the next chapter.

Example 1: Derivative of a Tangent Function
Example 2: Limit of a Cosine Function
Example 3: Angular Velocity

Home Trigonometric Functions Derivatives of Trigonometric Functions Theorem 3: Derivatives of Trigonometric Functions

Last Update: 2006-11-05