TRIGONOMETRIC FUNCTIONS

Continuity of Trigonometric Functions

Continuity of Trigonometric Functions

The SINE and COSINE functions are continuous at every real number.

Continuity of Trigonometric Functions

Continuity of Trigonometric Functions

The TANGENT, COTANGENT, SECANT and COSECANT functions are continuous over their respective domains.

Derivatives of SINE and COSINE

xcosxsinDx

xsinxcosDx

Derivative of TANGENT

xtanDx

xcosxsin

Dx

xcos

xcosDxsinxsinDxcos xx2

xcos

xsinxsinxcosxcos2

xcos

xsinxcos2

22 xcos2

1 xsec2

Derivative of COSECANT

xcscDx

xsinDx

1

xsin

xsinDDxsin xx211

xsin

xcosxsin2

0 xsin

xcos2

xsinxcos

xsin 1 xcotxcsc

Derivatives of Trigonometric Functions

xcosxsinDx xsinxcosDx

xsecxtanDx2

xcscxcotDx2

xtanxsecxsecDx xcotxcscxcscDx

Differentiability

The TANGENT, COTANGENT, SECANT and COSECANT functions are differentiable over their respective domains.

The SINE and COSINE functions are differentiable at every real number.

CHAIN RULE

Let u be a differentiable function of x.

uDucosusinD xx uDusinucosD xx

uDusecutanD xx 2

uDucscucotD xx 2

uDutanusecusecD xx uDucotucscucscD xx

Example 1. Evaluate

Solution:

xcosxsinDx

xsinDxcosxcosDxsin xx

xcosxcosxsinxsin

xcosxsin 22

xcosxsinDx

xsinxcos 22

Example 2. Evaluate

Solution: xsecxtanDx 235

xsecDxtanD xx 235

xDxsec x 335 2

xtanxsecxsec 222315 2

xsecxtanDx 235

xDxtanxsec x 222

Example 3. Evaluate

Solution:

xcsccotDx

xcscDxcsccsc x 2

xcotxcscxcsccsc 2

xcsccotDx

xcotxcscxcsccsc 2

xcsccscxcotxcsc 2

EXERCISE. Evaluate

xsec

xtanxDx 2

3

22 xcosxsinDx

xcotcscDx3

END