d Derivative of Trigonometric Functions i E 2 Lectures a l l - - PowerPoint PPT Presentation

▶
d
SMART_READER_LITE
LIVE PREVIEW

d Derivative of Trigonometric Functions i E 2 Lectures a l l - - PowerPoint PPT Presentation

Feb 13, 2024 387 likes •632 views

Section 3.5 d Derivative of Trigonometric Functions i E 2 Lectures a l l u d b Dr. Abdulla Eid A . College of Science r D MATHS 101: Calculus I Dr. Abdulla Eid (University of Bahrain) Trigonometric functions 1 / 23 d 1 Review

slide-1
SLIDE 1

D r . A b d u l l a E i d

Section 3.5 Derivative of Trigonometric Functions 2 Lectures

  • Dr. Abdulla Eid

College of Science

MATHS 101: Calculus I

  • Dr. Abdulla Eid (University of Bahrain)

Trigonometric functions 1 / 23

slide-2
SLIDE 2

D r . A b d u l l a E i d

1 Review of the trigonometric functions (Pre–Calculus). 2 Limits involving trigonometric functions. 3 Derivative of the basic trigonometric functions. 4 Derivative of the functions that involve trigonometric functions.

  • Dr. Abdulla Eid (University of Bahrain)

Trigonometric functions 2 / 23

slide-3
SLIDE 3

D r . A b d u l l a E i d

Review of the trigonometric functions (Pre–Calculus)

1 Radian and degree of an angle. 2 Definition of sine and cosine functions and their graphs. 3 Definition of the other trigonometric functions. 4 Some important trigonometric identities.

  • Dr. Abdulla Eid (University of Bahrain)

Trigonometric functions 3 / 23

slide-4
SLIDE 4

D r . A b d u l l a E i d

Radian and Degree

Angles are usually measured by their degree, for example, 30◦, 45◦, 90◦, 180◦, etc. On the other hand, angles as real numbers are given in terms of radian. degree → radian degree 180◦ · π radian → degree radian π · 180◦

  • Dr. Abdulla Eid (University of Bahrain)

Trigonometric functions 4 / 23

slide-5
SLIDE 5

D r . A b d u l l a E i d

degree → radian degree 180◦ · π radian → degree radian π · 180◦

Exercise 1

Fill in the following table Degree 0◦ 30◦ 60◦ 90◦ 120◦ 180◦ 270◦ 360◦ Radian

π 4 3π 4

  • Dr. Abdulla Eid (University of Bahrain)

Trigonometric functions 5 / 23

slide-6
SLIDE 6

D r . A b d u l l a E i d

Definition of Sine and Cosine

Unit Circe − 1 ≤ sin θ, cos θ ≤ 1 Right Triangle sin θ = opposite hypotunse cos θ = adjacent hypotunse

  • Dr. Abdulla Eid (University of Bahrain)

Trigonometric functions 6 / 23

slide-7
SLIDE 7

D r . A b d u l l a E i d

Example 2

Compute using the unit circle the values of sin π

2 , cos π 2 , sin π, cos π.

Solution: sin π 2 = cos π 2 = sin π = cos π =

  • Dr. Abdulla Eid (University of Bahrain)

Trigonometric functions 7 / 23

slide-8
SLIDE 8

D r . A b d u l l a E i d

Exercise 3

Fill in the following table using a calculator θ

π 6 π 4 π 3 π 2 2π 3 3π 4 5π 6

π

3π 2

2π sin θ cos θ

  • Dr. Abdulla Eid (University of Bahrain)

Trigonometric functions 8 / 23

slide-9
SLIDE 9

D r . A b d u l l a E i d

Graph of since and cosine

x y

  • Dr. Abdulla Eid (University of Bahrain)

Trigonometric functions 9 / 23

slide-10
SLIDE 10

D r . A b d u l l a E i d

Graph of since and cosine

x y

  • Dr. Abdulla Eid (University of Bahrain)

Trigonometric functions 10 / 23

slide-11
SLIDE 11

D r . A b d u l l a E i d

3 - Definition of the other trigonometric functions

tan θ = sin θ cos θ = opposite adjacent cot θ = cos θ sin θ = adjacent

  • pposite

sec θ = 1 cos θ = hypotunse adjacent csc θ = 1 sin θ = hypotunse

  • pposite

x y All> 0 sin θ > 0 tan θ > 0 cos θ > 0 “All Students Take Calculus“

  • Dr. Abdulla Eid (University of Bahrain)

Trigonometric functions 11 / 23

slide-12
SLIDE 12

D r . A b d u l l a E i d

Some useful trigonometric identities

1 sin2 θ + cos2 θ = 1. 2 1 + tan2 θ = sec2 θ

(1 + cot2 θ = csc2 θ).

3 sin(−θ) = − sin θ

(odd function) cos(−θ) = cos θ (even function).

4 Double angle formula:

sin(2θ) = 2 sin θ cos θ . cos(2θ) = cos2(θ) − sin2(θ) . cos2 θ = 1

2(1 + cos(2θ).

sin2 θ = 1

2(1 − cos(2θ).

5 sin(a + b) = sin a cos b + cos a sin b.

cos(a + b) = cos a cos b − sin a sin b. These are very useful formula!

  • Dr. Abdulla Eid (University of Bahrain)

Trigonometric functions 12 / 23

slide-13
SLIDE 13

D r . A b d u l l a E i d

Continuity of Sine and Cosine

Exercise 4

Prove that f is continuous at a if and only if lim

h→0 f (a + h) = f (a)

Exercise 5

Use the trigonometric identities to show that lim

h→0 sin(a + h) = sin(a)

and use the exercise above to show that f (x) = sin x is a continuous

  • function. Do the same for the f (x) = cos x.
  • Dr. Abdulla Eid (University of Bahrain)

Trigonometric functions 13 / 23

slide-14
SLIDE 14

D r . A b d u l l a E i d

2 - Limits involving trigonometric functions

Example 6

lim

θ→0

sin θ θ = 1 Solution: area of ≤ area of ≤ area of 1 2 sin θ ≤ 1 2θ ≤ 1 2 tan θ 1 ≤ θ sin θ ≤ tan θ sin θ 1 ≥ sin θ θ ≥ cos θ → lim

θ→0

sin θ θ = 1

  • Dr. Abdulla Eid (University of Bahrain)

Trigonometric functions 14 / 23

slide-15
SLIDE 15

D r . A b d u l l a E i d

Example 7

Find lim

θ→0

1 − cos θ θ Solution: lim

θ→0

1 − cos θ θ = lim

θ→0

1 − cos θ θ · 1 + cos θ 1 + cos θ = lim

θ→0

1 − cos2 θ θ(1 + cos θ) = lim

θ→0

sin2 θ θ(1 + cos θ) = lim

θ→0

sin θ θ · sin θ (1 + cos θ) = 1 · 0 = 0

  • Dr. Abdulla Eid (University of Bahrain)

Trigonometric functions 15 / 23

slide-16
SLIDE 16

D r . A b d u l l a E i d

Exercise 8

Find lim

θ→0

tan θ θ Solution: lim

θ→0

tan θ θ = lim

θ→0

sin θ θ cos θ = lim

θ→0

sin θ θ · 1 (cos θ) = 1 · 1 = 1

  • Dr. Abdulla Eid (University of Bahrain)

Trigonometric functions 16 / 23

slide-17
SLIDE 17

D r . A b d u l l a E i d

Three Important limits

lim

θ→0

sin θ θ = 1 lim

θ→0

1 − cos θ θ = 0 lim

θ→0

tan θ θ = 1

  • Dr. Abdulla Eid (University of Bahrain)

Trigonometric functions 17 / 23

slide-18
SLIDE 18

D r . A b d u l l a E i d

Example 9

Find lim

x→0

sin(7x) x Solution: lim

x→0

sin(7x) x = lim

x→0

7 sin(7x) 7x = 7 lim

x→0

sin(7x) 7x = 7

  • Dr. Abdulla Eid (University of Bahrain)

Trigonometric functions 18 / 23

slide-19
SLIDE 19

D r . A b d u l l a E i d

Example 10

Find lim

x→0 x sin

1 x

  • Solution:

− 1 ≤ sin 1 x

  • ≤ 1

− x ≤ x sin 1 x

  • ≤ x

lim

x→0 −x ≤ lim x→0 x sin

1 x

  • ≤ lim

x→0 x

0 ≤ lim

x→0 x sin

1 x

  • ≤ 0

lim

x→0 x sin

1 x

  • = 0
  • Dr. Abdulla Eid (University of Bahrain)

Trigonometric functions 19 / 23

slide-20
SLIDE 20

D r . A b d u l l a E i d

Exercise 11

Find lim

x→0+

√xesin( π

x )

Solution: − 1 ≤ sin π x

  • ≤ 1

e−1 ≤esin( π

x ) ≤ e1

− √xe−1 ≤√xesin( π

x ) ≤ √xe1

lim

x→0+ −√xe−1 ≤ lim x→0+

√xesin( π

x ) ≤ lim

x→0+

√xe1 0 ≤ lim

x→0+

√xesin( π

x ) ≤ 0

lim

x→0+

√xesin( π

x ) = 0

  • Dr. Abdulla Eid (University of Bahrain)

Trigonometric functions 20 / 23

slide-21
SLIDE 21

D r . A b d u l l a E i d

Example 12

Find lim

x→∞

sin (x) x2 + 1 Solution: − 1 ≤ sin (x) ≤ 1 0 ≤ sin2 (x) ≤ 1 0 ≤sin2 (x) x2 + 1 ≤ 1 x2 + 1 lim

x→∞ 0 ≤ lim x→∞

sin2 (x) x2 + 1 ≤ lim

x→∞

1 x2 + 1 0 ≤ lim

x→∞

sin2 (x) x2 + 1 ≤ 0 lim

x→∞

sin (x) x2 + 1 = 0

  • Dr. Abdulla Eid (University of Bahrain)

Trigonometric functions 21 / 23

slide-22
SLIDE 22

D r . A b d u l l a E i d

Example 13

For which value(s) of k is the function defined by f (x) =

  • sin x

x ,

x < 0 2e3x − k, x ≥ 0 continuous at x = 0? Solution: We need to compute the left and right limit and we make them equal. lim

x→0− f (x) = lim x→0−

sin x x = 1 lim

x→0+ f (x) = lim x→0+ 2e3x − k = 2 − k

we have 1 = 2 − k → k = 1.

  • Dr. Abdulla Eid (University of Bahrain)

Trigonometric functions 22 / 23

slide-23
SLIDE 23

D r . A b d u l l a E i d

Exercise 14

For which value(s) of k is the function defined by f (x) = sin(2x)

x

, x < 0 cos x + x2 + 4k, x ≥ 0 continuous at x = 0?

  • Dr. Abdulla Eid (University of Bahrain)

Trigonometric functions 23 / 23