EMAT 1173 Fundamental Mathematics I Lecture Notes (PDF)

Shared using Xodo PDF Reader and Editor EMAT 1173 - Fundamental Mathematics I Sanjeewa Karunarathna Department of Mathematical Sciences Wayamba University of Sri Lanka...

Shared using Xodo PDF Reader and Editor EMAT 1173 - Fundamental Mathematics I Sanjeewa Karunarathna Department of Mathematical Sciences Wayamba University of Sri Lanka Kuliyapitiya [email protected], [email protected] July 09, 2024 Sanjeewa Karunarathna (WUSL) WEEK 02 July 09, 2024 1 / 25  Shared using Xodo PDF Reader and Editor Chapter 1: Trigonometry Summary from WEEK 01 ♡ Definition of angle... ♡ Positive and negative angles... ♡ Degrees and radians... ♡ Relationship between degrees, minutes and seconds... ♡ Convert angles in decimal forms to degrees, minutes and seconds and vise versa... Example Convert 30◦ 15′ 18” to a decimal in degrees. Sol: Ans: 30.255◦ Example Convert 156.742◦ to degrees, minutes and seconds form. Sol: Ans: 156◦ 44′ 31.2′′ Sanjeewa Karunarathna (WUSL) WEEK 02 July 09, 2024 2 / 25  Shared using Xodo PDF Reader and Editor Chapter 1: Trigonometry Summary from WEEK 01 (cont.) ♡ Relationship between degrees and radians... ♡ Convert angles in degrees to radians and vise versa... Example Convert each degree measure into radians. 1. 345◦ 2. −510◦ Sol: Example Convert each radian measure into degrees. 1. 35π 18 2. − 41π 36 Sol: Sanjeewa Karunarathna (WUSL) WEEK 02 July 09, 2024 3 / 25  Shared using Xodo PDF Reader and Editor Chapter 1: Trigonometry Summary from WEEK 01 (cont.) ♡ The theorem of Pythagoras (Pythagorean theorem)... Example Does this triangle have a right angle? Sol: Sanjeewa Karunarathna (WUSL) WEEK 02 July 09, 2024 4 / 25  Shared using Xodo PDF Reader and Editor Chapter 1: Trigonometry ♡ Trigonometric functions... Example Consider the following right-angled triangle: Find the following trigonometric functions for the above triangle. (i) sin θ (ii) cos θ (iii) tan θ (iv) csc θ (v) sec θ (vi) cot θ Sol: Sanjeewa Karunarathna (WUSL) WEEK 02 July 09, 2024 5 / 25  Shared using Xodo PDF Reader and Editor Chapter 1: Trigonometry ♡ Signs of trigonometric functions in each quadrant... Example What is the sign of (i) cos(165◦ ) (ii) sin(245◦ ) (iii) tan(300◦ ) (iv) sin( 2π 3 ) (v) cos( 4π 3 ) (vi) tan( π 3 ) Sol: Sanjeewa Karunarathna (WUSL) WEEK 02 July 09, 2024 6 / 25  Shared using Xodo PDF Reader and Editor Chapter 1: Trigonometry ♡ The following memory device can be used to memorize signs of trigonometric functions in each quadrant: ♡ Co-function identities... Negative angle identities ♡ They can be used as formulas when trigonometric functions appear with negative angles. ♣ The negative angle identities are helpful to transform any trigonometric function which contains negative angle as same trigonometric ratio with positive angle. sin(−θ) = − sin(θ); cos(−θ) Sanjeewa Karunarathna (WUSL) WEEK − 09, =02cos(θ); tan(−θ) =July tan2024θ 7 / 25  Shared using Xodo PDF Reader and Editor Chapter 1: Trigonometry Example Evaluate the cosecant function csc( 5π 6 ). Sol: First, we may transform csc to basic equivalent sin : 1   5π csc =. 6 sin( 5π 6 ) Next, we evaluate sin( 5π6 ). Apply the cofunction identity for sine: sin( 6 ) = sin( 2 + 3 ) = sin( π2 − ( −π 5π π π −π 3 )) = cos( 3 ). Using negative angle identy,     −π π cos = cos. 3 3 As cos( π3 ) = 12 , sin( 5π 1 5π 6 ) = 2. Therefore, csc( 6 ) = 2 since 1   csc 5π6 =. sin( 5π6 ) Sanjeewa Karunarathna (WUSL) WEEK 02 July 09, 2024 8 / 25  Shared using Xodo PDF Reader and Editor Chapter 1: Trigonometry Example Using co-function identies, evaluate the followings:         2π 2π 3π 5π (i) cos (ii) sin (iii) cot (iv) sec. 3 3 4 6 Sol: LTS Supplement angle identities ♡ Identities expressing trig functions in terms of their supplements. sin(180◦ − θ) = sin(θ); cos(180◦ − θ) = − cos(θ); tan(180◦ − θ) = − tan θ Example Sanjeewa Karunarathna (WUSL) WEEK 02 July 09, 2024 9 / 25  Shared using Xodo PDF Reader and Editor Chapter 1: Trigonometry Graphs of trigonometric functions ♡ Graph of sine function Sanjeewa Karunarathna (WUSL) WEEK 02 July 09, 2024 10 / 25  Shared using Xodo PDF Reader and Editor Chapter 1: Trigonometry Graphs of trigonometric functions (Cont.) ♡ Graph of cosine function Sanjeewa Karunarathna (WUSL) WEEK 02 July 09, 2024 11 / 25  Shared using Xodo PDF Reader and Editor Chapter 1: Trigonometry Graphs of trigonometric functions (Cont.) ♡ Graph of tangent function Sanjeewa Karunarathna (WUSL) WEEK 02 July 09, 2024 12 / 25  Shared using Xodo PDF Reader and Editor Chapter 1: Trigonometry Pythagorean relationships sin2 θ + cos2 θ = 1 1 + tan2 θ = sec2 θ 1 + cot2 θ = csc2 θ Example Using the basic relationships, find the values of the trigonometric functions of θ for given sin θ = 54. Sol: Values of sin θ are positive in the first and second q quadrant. As we p have sin θ + cos θ = 1, cos θ = ± 1 − sin θ = ± 1 − ( 45 )2. Hence, 2 2 2 r 9 3 cos θ = ± =±. 25 5 Sanjeewa Karunarathna (WUSL) WEEK 02 July 09, 2024 13 / 25  Shared using Xodo PDF Reader and Editor Chapter 1: Trigonometry First quadrant As cos is positive in the first quadrant, 3 sin θ = 54 ; cos θ = 35 ; tan θ = cos sin θ 4 5 5 θ = 3 ; csc θ = 4 ; sec θ = 3 ; cot θ = 4. Second quadrant sin θ = 54 ; cos θ = − 35 ; tan θ = cos sin θ 4 5 5 θ = − 3 ; csc θ = 4 ; sec θ = − 3 ; cot θ = 3 −. 4 Example Use Pythagorean identities to find missing trigonometric values. If sec u = − 32 and tan u > 0, find the value of the other trig functions. Sanjeewa Karunarathna (WUSL) WEEK 02 July 09, 2024 14 / 25  Shared using Xodo PDF Reader and Editor Chapter 1: Trigonometry Sol: 1 As sec u = cosu , cos u = − 23. Using Pythagorean identity sin2 u + cos2 u = 1, we have √ 5 sin u = ±. 3 As cos u 0, u is in Quadrant III which means sin u = − 35. Hence, LTS Example If cot(x) = −5/12, x lies in the fourth quadrant, then find the value of csc(x). 13 Sol: Ans: −. 12 Example √ Sanjeewa Karunarathna (WUSL) WEEK 02 3 July 09, 2024 15 / 25  Shared using Xodo PDF Reader and Editor Chapter 1: Trigonometry Example Use Pythagorean identities to simplify trigonometric expressions. Simplify sin x cos2 x − sin x Sol: − sin3 x Definition Addition formulas sin(α + β) = sin α cos β + cos α sin β, cos(α + β) = cos α cos β − sin α sin β, tan α + tan β tan(α + β) = (1 − tan α tan β) Example Find sin(75◦ ), cos(75◦ ) and tan(75◦ ) Sanjeewa Karunarathna (WUSL) WEEK 02 July 09, 2024 16 / 25  Shared using Xodo PDF Reader and Editor Chapter 1: Trigonometry Sol: As 75◦ = 45◦ + 30◦ , sin(75◦ ) = sin(45◦ + 30◦ ).Using addition formula of sin, sin(45◦ + 30◦ ) = sin(45◦ ◦ √ ) cos(30 ) + cos(45 √ ◦ ) sin(30◦ ) 1 3 1 1 ( 3 + 1) =√ +√ = √ 2 2 22 2 2 Consider, cos(75◦ ) : LTS Consider, tan(75◦ ) : LTS Example Find sin(120◦ ), cos(120◦ ) and tan( 2π 3 ) Sol: LTS Example 3π 3π 3π


Find sin 4 , cos 4 and tan 4 Sol: LTS Sanjeewa Karunarathna (WUSL) WEEK 02 July 09, 2024 17 / 25  Shared using Xodo PDF Reader and Editor Chapter 1: Trigonometry Example Find sin( 5π 5π 5π 6 ), cos( 6 ) and tan( 6 ) Sol: LTS Definition Subtraction formulas sin(α − β) = sin α cos β − cos α sin β, cos(α − β) = cos α cos β + sin α sin β, tan α − tan β tan(α − β) = (1 + tan α tan β) Example π π π Find sin( 12 ), cos( 12 ) and tan( 12 ) π π Sol: We have 12 = 4 − π6. Therefore, Sanjeewa Karunarathna (WUSL) WEEK 02 July 09, 2024 18 / 25  Shared using Xodo PDF Reader and Editor Chapter 1: Trigonometry Summary from WEEK 02 ♡ Co-function identities... ♡ Negative angle identities... ♡ Supplement angle identities... Example Find the exact value of each expression. i. cos(−240◦ ) ii. tan(675◦ ). 1 Sol: i. − ii. − 1 2 Example Find the values of the six trigonometric functions for 210◦ Sol: Sanjeewa Karunarathna (WUSL) WEEK 02 July 09, 2024 19 / 25  Shared using Xodo PDF Reader and Editor Chapter 1: Trigonometry Sol (Cont.): π π π  cos = cos − 12 4 6 π  π  π  π  = cos cos + sin sin 4 6 4 6 √ √ 1 3 1 1 ( 3 + 1) =√ +√ = √ 2 2 2 2 2 2 π Consider, sin( 12 ): LTS π Consider, tan( 12 ): LTS Example Find sin( 7π 7π 7π 12 ), cos( 12 ) and tan( 12 ) Sol: LTS Sanjeewa Karunarathna (WUSL) WEEK 02 July 09, 2024 20 / 25  Shared using Xodo PDF Reader and Editor Chapter 1: Trigonometry Example Suppose we know that sin A = 35 and that cos B = 5 13 , where A and B are acute angles. Find sin(A + B) and cos(A − B). Sol: LTS Definition Double-angle formulas sin 2α = 2 sin α cos α cos 2α = cos2 α − sin2 α or cos 2α = 1 − 2 sin2 α or cos 2α = 2 cos2 α − 1 2 tan α tan 2α = 1 − tan2 α Sanjeewa Karunarathna (WUSL) WEEK 02 July 09, 2024 21 / 25  Shared using Xodo PDF Reader and Editor Chapter 1: Trigonometry Example Using addition formulas and double-angle formulas, find sin 3x in terms of sin x. Sol: As 3x = 2x + x, sin 3x = sin(2x + x) LTS Example Using addition formulas and double-angle formulas, find cos 3x in terms of cos x. Sol: LTS Example Verify the three double-angle formulas (sin 2A, cos 2A, tan 2A) for the cases π A = 30◦ and A =. 4 Sol: LTS Sanjeewa Karunarathna (WUSL) WEEK 02 July 09, 2024 22 / 25  Shared using Xodo PDF Reader and Editor Chapter 1: Trigonometry Example 3 If sin A = 5 with A is in the 2nd quadrant, find sin 2A. Sol: To apply the formula for sin 2A, we must first find cosA. p cos A = ± 1 − sin2 A s  2 3 =− 1− , 5 because A terminates in QII, cos A is negative. Then cos A = − 45. sin 2A = 2 sin A cos A    3 4 =2 − 5 5 24 =− 25. Sanjeewa Karunarathna (WUSL) WEEK 02 July 09, 2024 23 / 25  Shared using Xodo PDF Reader and Editor Chapter 1: Trigonometry Example Prove (sin θ + cos θ)2 = 1 + sin 2θ. Sol: LTS Example 2 cot x Prove sin 2x =. 1 + cot 2 x Sol: LTS Example If sin A = √1 , find cos 2A. 5 Sol: Ans. ( 35 ) Sanjeewa Karunarathna (WUSL) WEEK 02 July 09, 2024 24 / 25  Shared using Xodo PDF Reader and Editor Chapter 1: Trigonometry THANK YOU Be Strong! Sanjeewa Karunarathna (WUSL) WEEK 02 July 09, 2024 25 / 25