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

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 of the trigonometric functions (Pre–Calculus). i E 2 Limits involving trigonometric functions. a l l 3 Derivative of the basic trigonometric functions. u d 4 Derivative of the functions that involve trigonometric functions. b A . r D Dr. Abdulla Eid (University of Bahrain) Trigonometric functions 2 / 23

Review of the trigonometric functions (Pre–Calculus) d i E 1 Radian and degree of an angle. a l 2 Definition of sine and cosine functions and their graphs. l u d 3 Definition of the other trigonometric functions. b A 4 Some important trigonometric identities. . r D Dr. Abdulla Eid (University of Bahrain) Trigonometric functions 3 / 23

Radian and Degree Angles are usually measured by their degree, for example, 30 ◦ , 45 ◦ , d i 90 ◦ , 180 ◦ , etc. E On the other hand, angles as real numbers are given in terms of a l l radian. u d b A degree → radian radian → degree . degree radian r · 180 ◦ 180 ◦ · π D π Dr. Abdulla Eid (University of Bahrain) Trigonometric functions 4 / 23

degree → radian radian → degree degree d radian · 180 ◦ 180 ◦ · π i E π a l l u Exercise 1 d b Fill in the following table A . 0 ◦ 30 ◦ 60 ◦ 90 ◦ 120 ◦ 180 ◦ 270 ◦ 360 ◦ Degree r D π 3 π Radian 4 4 Dr. Abdulla Eid (University of Bahrain) Trigonometric functions 5 / 23

Definition of Sine and Cosine Right Triangle Unit Circe d i E a l l u d b A . r D sin θ = opposite hypotunse cos θ = adjacent − 1 ≤ sin θ , cos θ ≤ 1 hypotunse Dr. Abdulla Eid (University of Bahrain) Trigonometric functions 6 / 23

Example 2 Compute using the unit circle the values of sin π 2 , cos π 2 , sin π , cos π . d Solution: i E a l sin π l u 2 = d b cos π 2 = A . sin π = r D cos π = Dr. Abdulla Eid (University of Bahrain) Trigonometric functions 7 / 23

d Exercise 3 i E Fill in the following table using a calculator a l π π π π 2 π l 3 π 5 π 3 π 0 2 π θ π u 6 4 3 2 3 4 6 2 d sin θ b cos θ A . r D Dr. Abdulla Eid (University of Bahrain) Trigonometric functions 8 / 23

Graph of since and cosine y d i E a x l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Trigonometric functions 9 / 23

Graph of since and cosine y d i E a x l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Trigonometric functions 10 / 23

3 - Definition of the other trigonometric functions tan θ = sin θ cos θ = opposite cot θ = cos θ sin θ = adjacent adjacent opposite d i E cos θ = hypotunse 1 sin θ = hypotunse 1 sec θ = csc θ = a adjacent opposite l l u y d b A sin θ > 0 All > 0 . r D x tan θ > 0 cos θ > 0 “ A ll S tudents T ake C alculus“ Dr. Abdulla Eid (University of Bahrain) Trigonometric functions 11 / 23

Some useful trigonometric identities 1 sin 2 θ + cos 2 θ = 1. 2 1 + tan 2 θ = sec 2 θ ( 1 + cot 2 θ = csc 2 θ ) . d i E 3 sin ( − θ ) = − sin θ ( odd function ) a cos ( − θ ) = cos θ ( even function ) . l l u d 4 Double angle formula: b A sin ( 2 θ ) = 2 sin θ cos θ . cos ( 2 θ ) = cos 2 ( θ ) − sin 2 ( θ ) . . r cos 2 θ = 1 D 2 ( 1 + cos ( 2 θ ) . sin 2 θ = 1 2 ( 1 − cos ( 2 θ ) . 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

Continuity of Sine and Cosine Exercise 4 Prove that f is continuous at a if and only if d h → 0 f ( a + h ) = f ( a ) lim i E a l l u d b A Exercise 5 Use the trigonometric identities to show that . r D h → 0 sin ( a + h ) = sin ( a ) lim 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

2 - Limits involving trigonometric functions Example 6 sin θ lim = 1 θ θ → 0 d i Solution: E a l l u d b A area of ≤ area of ≤ area of . r D 1 2 sin θ ≤ 1 2 θ ≤ 1 2 tan θ sin θ ≤ tan θ θ 1 ≤ sin θ 1 ≥ sin θ sin θ = 1 ≥ cos θ → lim θ θ θ → 0 Dr. Abdulla Eid (University of Bahrain) Trigonometric functions 14 / 23

Example 7 Find 1 − cos θ lim θ θ → 0 Solution: d i E 1 − cos θ 1 − cos θ a · 1 + cos θ = lim lim l l 1 + cos θ θ u θ θ → 0 θ → 0 d 1 − cos 2 θ b = lim A θ ( 1 + cos θ ) θ → 0 sin 2 θ . r = lim D θ ( 1 + cos θ ) θ → 0 sin θ sin θ = lim · ( 1 + cos θ ) θ θ → 0 = 1 · 0 = 0 Dr. Abdulla Eid (University of Bahrain) Trigonometric functions 15 / 23

Exercise 8 Find tan θ lim θ d θ → 0 i E Solution: a l l u d tan θ sin θ = lim b lim θ cos θ θ A θ → 0 θ → 0 sin θ 1 . = lim · r θ ( cos θ ) D θ → 0 = 1 · 1 = 1 Dr. Abdulla Eid (University of Bahrain) Trigonometric functions 16 / 23

Three Important limits d i E sin θ a lim = 1 l θ θ → 0 l u 1 − cos θ d lim = 0 b θ θ → 0 A tan θ = 1 lim . r θ θ → 0 D Dr. Abdulla Eid (University of Bahrain) Trigonometric functions 17 / 23

Example 9 Find sin ( 7 x ) lim d x x → 0 i E a Solution: l l u d b sin ( 7 x ) 7 sin ( 7 x ) lim A = lim x 7 x x → 0 x → 0 . sin ( 7 x ) r D = 7 lim 7 x x → 0 = 7 Dr. Abdulla Eid (University of Bahrain) Trigonometric functions 18 / 23

Example 10 Find � 1 � x → 0 x sin lim x d Solution: i E a � 1 � l − 1 ≤ sin ≤ 1 l u x d b � 1 � − x ≤ x sin A ≤ x x . � 1 � r D x → 0 − x ≤ lim lim x → 0 x sin ≤ lim x → 0 x x � 1 � 0 ≤ lim x → 0 x sin ≤ 0 x � 1 � = 0 x → 0 x sin lim x Dr. Abdulla Eid (University of Bahrain) Trigonometric functions 19 / 23

Exercise 11 Find √ xe sin ( π x ) lim x → 0 + Solution: d i E a � π � l − 1 ≤ sin ≤ 1 l u x d e − 1 ≤ e sin ( π x ) ≤ e 1 b A − √ xe − 1 ≤√ xe sin ( π x ) ≤ √ xe 1 . r x → 0 + −√ xe − 1 ≤ lim √ xe sin ( π √ xe 1 D x ) ≤ lim lim x → 0 + x → 0 + √ xe sin ( π x ) ≤ 0 0 ≤ lim x → 0 + √ xe sin ( π x ) = 0 lim x → 0 + Dr. Abdulla Eid (University of Bahrain) Trigonometric functions 20 / 23

Example 12 Find sin ( x ) lim x 2 + 1 x → ∞ Solution: d i E a − 1 ≤ sin ( x ) ≤ 1 l l u 0 ≤ sin 2 ( x ) ≤ 1 d b 0 ≤ sin 2 ( x ) 1 A x 2 + 1 ≤ x 2 + 1 . r sin 2 ( x ) 1 D x → ∞ 0 ≤ lim lim x 2 + 1 ≤ lim x 2 + 1 x → ∞ x → ∞ sin 2 ( x ) 0 ≤ lim x 2 + 1 ≤ 0 x → ∞ sin ( x ) lim x 2 + 1 = 0 x → ∞ Dr. Abdulla Eid (University of Bahrain) Trigonometric functions 21 / 23

Example 13 For which value(s) of k is the function defined by � sin x x , x < 0 d f ( x ) = i 2 e 3 x − k , E x ≥ 0 a l continuous at x = 0? l u d Solution: We need to compute the left and right limit and we make them b A equal. . r sin x D x → 0 − f ( x ) = lim = 1 lim x x → 0 − x → 0 + 2 e 3 x − k = 2 − k x → 0 + f ( x ) = lim lim we have 1 = 2 − k → k = 1. Dr. Abdulla Eid (University of Bahrain) Trigonometric functions 22 / 23

Exercise 14 For which value(s) of k is the function defined by � sin ( 2 x ) , x < 0 f ( x ) = x cos x + x 2 + 4 k , x ≥ 0 d i E continuous at x = 0? a l l u d b A . r D Dr. Abdulla Eid (University of Bahrain) Trigonometric functions 23 / 23