EMAT 1173 Fundamental Mathematics I Lecture Notes (PDF)
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Uploaded by FresherMaclaurin6046
Wayamba University of Sri Lanka
2024
Sanjeewa Karunarathna
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Summary
These lecture notes provide a summary of fundamental mathematics (EMAT 1173), focusing on trigonometry. The document covers topics like angles, degrees, radians, and trigonometric functions. It includes examples and exercises.
Full Transcript
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