Trigonometry Topis overed: Pythgors Theorem Definition of sin, os nd tn Solving right-ngle tringles Sine nd osine rule Lelling right-ngle tringle Opposite (Side opposite the ngle θ) Hypotenuse (Side opposite the right-ngle is the longest side) djent (Side djent to the ngle θ) θ Pythgors Theorem If the lengths of two sides of right-ngle tringle re known, then Pythgors Theorem n e used to find the length of the missing side. The Theorem sttes tht: = + 1
Exmple 1: Find the length of the missing side: 3m 6 m Solution: Lelling the tringle: () 3m () 6 m () Using Pythgors Theorem: = + = (3) + (6) = 9 + 36 = 45 = 45 = 6.708 Therefore the length of side is 6.7m (rounded to 1 deiml ple).
Exmple : Find the length of the missing side: 4 m 10m Solution: Lelling the tringle: 4 m () 10m () () Using Pythgors Theorem: = + (10) = (4) + 100 = 16 + = 100 16 = 84 = 84 = 9.165 Therefore the length of side is 9.m (rounded to 1 deiml ple). 3
Questions Find the missing side of the following two tringles (giving your nswers to 1 deiml ple).. m 5 m. 1 m 4 m Solutions:. m () () 5 m () Using Pythgors Theorem: = + = () + (5) = 4 + 5 = 9 = 9 = 5.385 Therefore the length of side is 5.4m (rounded to 1 deiml ple). Using Pythgors Theorem:. () 1 m () = + (1) = + (4) 4 m () 144 = + 16 = 144-16 = 18 = 18 = 11.3137 Therefore the length of side is 11.3m (rounded to 1 deiml ple). 4
Definition of sin, os nd tn opposite hypotenuse djent θ θ opp sin θ = hyp dj os θ = hyp opp tn θ = dj Exmple 1: Find sin θ, os θ nd tn θ of the following right-ngle tringle. 3 m θ 5 m Solution: Firstly need to lulte length of hypotenuse. Using Pythgors Theorem: = + = 3 + 5 34 = 9 + 5 3 m = 34 θ 5 m = 34 opp sin θ = hyp dj os θ = hyp opp tn θ = dj sin θ = 3 34 os θ = 5 34 tn θ = 5 3 θ = sin 1 3 34 = 31.0 o θ = os 1 5 34 = 31.0 o 3 θ = tn 1 = 31.0 o 5 5
Exmple : Find the length of the side mrked. 1 m m Solution: We know: hyp = 1 ngle θ = 3.5 o Missing side is opposite the ngle, so we use: opp sin θ = hyp sin (3.5) = 1 = 1sin(3.5) = 6.44759 Therefore the length of the side mrked is 6.45m (rounded to deiml ples). 3.5 o Exmple 3: Find the length of the side mrked y. y m Solution: We know: djent = 0 ngle θ = 4 o Missing side is hypotenuse, so we use: dj os θ = hyp 0 m 4 o 0 os (4) = y yos(4) = 0 y = 0 os( 4) = 6.9165 Therefore the length of the side mrked y is 6.91m (rounded to deiml ples). 6
Exmple 4: Find the length of the side mrked x. x m 10 m Solution: We know: the ngle θ = 50 o side djent to the ngle = 10 Missing side is opposite the ngle, so we use: opp tn θ = dj 50 o x tn(50) = 10 x = 10tn(50) x = 11.91753593 Therefore the length of the side mrked x is 11.9m (rounded to deiml ples). 7
Questions (give your nswers to deiml ples) 1. Find the length of the missing side mrked. 7 m m 4 o. Find the ngle in the following tringle: 10 m 8 m θ 3. Find the length of the side mrked x. 15 m 36 o x m 8
Solutions: 1. 7 m m opp sin θ = hyp sin(4) = 7 4 o sin(4) = 7 7 = sin(4) = 10.4613 = 10.46 (rounded to deiml ples). 8 m 10 m θ opp tn θ = dj tn θ = 10 8 θ = 8 tn 1 10 θ = 38.6598 θ = 38.66 (rounded to deiml ples) 3. 15 m 36 o x m dj os θ = hyp os(36) = 15 x x = 15os(36) = 1.1355 x = 1.14 (rounded to deiml ples) 9
Sine rule sin = sin = sin Use Sine rule when: Given 1 ngle nd sides nd to need to find nother ngle OR Given ngles nd 1 side nd need to find nother side. 10
Exmple 1: Find the length of the missing side lelled x. 7 m x 6 o 47 o Solution: Lelling the tringle: 7 m x We know, ngle = 47 o ngle = 6 o side = 7 Unknown side is. Therefore using 6 o 47 o = sin 7 sin47 = sin x sin6 7sin6 x = = 8.4509 sin47 Therefore the length of the side lelled x is 8.45m (rounded to deiml ples). 11
Exmple : Find the ngle lelled θ. 4 m θ 5 m 40 o Solution: Lelling the tringle: 4 m θ 5 m 40 o We know, ngle = 40 o side = 5 side = 4 Unknown ngle is. Therefore using = sin 5 = sinθ sin 4 sin40 4sinθ 5 = sin40 5sin(40) = 4sin(θ) sin(θ) = 5sin40 4 5sin40 θ = sin 1 4 = 53.464149.. Therefore the ngle is 53.46 o (rounded to deiml ples). 1
Questions 1. Find the length lelled x in the following tringle. 0m 6 o x m 50 o. Find the ngle mrked θ in the following tringle. 6 8 39 o θ Solutions 1. Lelling the tringle. 0m 6 o Unknown side x is side. Therefore using = sin x sin6 = sin 0 sin50 x m 0sin6 x = = 3.05148 sin50 50 o Therefore the length of x is 3.05 o (rounded to deiml ples). 13
. 6 8 39 o θ Using the sine rule: = sin 8 sin39 = sin 6 sinθ 8sinθ = sin39 6 8sinθ = 6sin39 6sin39 sinθ = 8 sinθ = 0.47199 θ = 8.16 o (rounded to deiml ples) 14
osine Rule Use the osine rule when given the length of sides nd the ngle etween them, you n then find the length of the missing side: = + os = + os = + os lso use the osine rule when given the lengths of ll three sides of tringle nd wnt to find the size of one of the ngles. os = os = os = + + + 15
Exmple 1 Find the length of the side. 7 m 71 o 9 m Solution 7 m 71 o 9 m = + os = 9 + 7 (9)(7)os(71) = 81 + 49 16os(71) = 130 16os(71) = 88.978 = 9.43m (rounded to deiml ples) 16
Exmple lulte the size of ngle. 3.6 m 5. m 4.3 m Solution 3.6 m 5. m 4.3 m os = os = + ( 5.) + ( 3.6) ( 4.3) ( 5.)( 3.6) 40 18.49 os = 37.44 os = 1.51 37.44 = 54.93 o (rounded to deiml ples) 17
Questions 1. Find the length of. 19 m 1 m 48 o. Find the length of the missing side nd the ngles t nd. m 55 o 1m 18
Solutions 1. Lelling the tringle. 19 m 1 m 48 o = + os = (19) + (1) (19)(1)os(48) = 361 + 144 456os(48) = 505 305.13 = 199.876 = 14.14 m (rounded to deiml ples). Missing side (): = + os m 55 o 1m = (1) + () (1)()os(55) = 144 + 484 58os(55) = 68 58os(55) = 35.15164 = 18.03196 Therefore the length of the missing side is 18.03m (rounded to deiml ples). ngle : os = os = + ( 18.03) + ( ) ( 1) ( 18.03)( ) 35.0809 + 484 144 os = 793.3 os = 0.838351 = 33.03356 Therefore ngle t is 33.03 o (rounded to deiml ples). 19
ngle : os = os = + ( 18.03) + ( 1) ( ) ( 18.03)( 1) 35.0809 + 144 484 os = 43.7 os = -0.034477 = 91.97580 Therefore ngle t is 91.98 o (rounded to deiml ples). 0