C2 Right Angled Trigonometry Ojetive: To know nd e le to use trigonometri rtios in rightngled tringles
opposite C Definition Trigonometry ws developed s method of mesuring ngles without ngulr units suh s degrees or rdins, involving right-ngled tringles. There re three min trigonometri rtios, nd severl derived funtions of these. A djent B The Sine funtion (or sin for short) is the rtio of the side opposite the ngle in question, to the hypotenuse. The Cosine (os) of n ngle is the rtio of the djent side to the ngle, to the hypotenuse. The Tngent (tn) is the rtio of the opposite to the djent.
opposite C Definition sin A opp. hyp. osa dj. hyp. SOH CAH TOA A djent B tna opp. dj.
opposite 15 Eg. Find the Sine, Cosine nd Tngent rtios for ngle A. A djent 8
Find the following: 1. sin A, given = 7, = 25, = 24 2. os A, given = 10, = 26, = 24 3. tn A, given = 33, = 183, = 180 4. os C, given = 24, = 30, = 18 5. sin C, given = 13, = 85, = 84 6. tn C, given = 48, = 73, = 55 C Solutions A B 1. sin A = 0.28 2. os A = 0.92 3. tn A = 0.18 4. os C = 0.8 5. sin C = 0.99 6. tn C = 1.15
opposite C More Definitions 1 ose A sin A The reiprol of eh rtio n lso e found nd used. A djent B 1 se A os A 1 ota tn A The reiprol of sine is osent. The reiprol of osine is sent. The reiprol of tngent is otngent. Every ngle hs relted sin, os, tn, ose, se nd ot, regrdless of the size of the tringle. Originlly these rtios were lulted nd reorded in tles for use. Now, however, our lultors store the informtion for us.
Find the following: 1. se C, given = 7, = 25, = 24 2. ose A, given = 10, = 26, = 24 3. ot A, given = 33, = 183, = 180 4. ose C, given = 24, = 30, = 18 5. ot C, given = 13, = 85, = 84 6. se C, given = 96, = 146, = 110 C Solutions A B 1. se C = 3.57 2. ose A = 2.6 3. ot A = 5.45 4. ose C = 1.67 5. ot C = 0.15 6. se C = 1.52
Eg. Find the lengths of sides nd. 32
Eg. Find the lengths of sides x nd y. x y 59
Find the missing two sides of these tringles: 1. A = 56, = 8 m 2. A = 26, = 11 m 3. C = 80, = 9 m 4. C = 67, = 43 m 5. A = 12.5, = 13 m 6. A = 79, = 12 m C Solutions A B 1. = 9.65 m; = 5.40 m 2. = 5.37 m; = 12.24 m 3. = 1.59 m; = 9.14 m 4. = 110.05 m; = 101.3 m 5. = 2.81 m; = 12.69 m 6. = 61.73 m; = 62.89 m
Eg. A surveyor, P, is 1000 m wy from the foot of rdio mst, Q, on horizontl ground. From P the ngle of elevtion of the top, R, of the mst is 20. Find the height of the mst. Assume the surveyor s height to e negligile.
Eg. From the top of vertil liff, 100m ove selevel, the ngle of depression of ot out t se is 32. How fr is the ot from the foot of the liff?
Tsk: Find the size of ngle A. 6 m A
opposite C Definitions The inverses of sin, os nd tn must e used in order to work kwrds to find the ngle. A sin 1 A os 1 A djent B A tn 1
Eg. A ot is 300 ft wy from n oserver t selevel, nd hs 30 ft mst. At wht ngle to the horizontl must the oserver look to see the top of the mst?
Find the required ngles of these tringles: 1. A, given = 8, = 12 2. A, given = 7, = 19 3. A, given = 9, = 4.5 4. C, given = 12, = 15 5. C, given = 1.6, = 3.2 6. C, given = 11, = 5.3 C Solutions A B 1. A = 41.8: 2. A = 20.22: 3. A = 60: 4. C = 51.34: 5. C = 60: 6. C = 28.80:
B It is lso possile to pply trigonometry to non-right-ngled tringles. A 24 Eg. Find the length of BC in the isoseles tringle ABC. C
Eg. A pyrmid is 45m tll, nd 55m long eh side of the se. Wht is the ngle etween the se nd the side?
θ Eg. Find the osine, sine nd tngent of ngle θ in this equilterl tringle, without using lultor.
Eg. Now find sin, os nd tn for eh of the speil ngles elow. Rtio 0 30 45 60 90 Sin Cos Tn 3 2 1 2 3 You re expeted to rememer the ove rtios. They re useful in skething grphs nd estimting other rtios.
The SINE Rule The sine rule llows us to use trigonometry to find sides or ngles of ny tringle, right-ngled or not. Proof C x A B
The SINE Rule sin A sinb sinc C x A B
Eg Find the missing sides nd ngles: 59 5.4 m 48 35 70 D 44 67 C
A-Level Pst Pper Question WJEC C2 - June 2009
The Amiguous Cse There re osions when the ngles found using the Sine Rule re miguous (they n e more thn one different ngle). This sitution rises from the sine urve s repetitive nture. 40 B B
Eg Find oth ses of these miguous missing ngles: F 5.4 m 48 65 E H 37 58 G
Tsk: Copy out the following tles of vlues nd, using your lultor, find the sin, os nd tn rtios for eh ngle. Drw grph of eh funtion on seprte set of xes. For sin x nd os x use 0 x 360 ; -1 y +1 For tn x use 0 x 360 ; -4 y +4 x 0 15 30 45 60 75 90 105 120 135 150 165 180 195 210 225 240 255 270 285 300 315 330 345 360 sin x os x tn x
1 y 0.8 0.6 0.4 0.2 30 60 90 120 150 180 210 240 270 300 330 360 390 420 450 480 510 540 x 0.2 0.4 0.6 0.8 1
1 y 0.8 0.6 0.4 0.2 30 60 90 120 150 180 210 240 270 300 330 360 390 420 450 480 510 540 x 0.2 0.4 0.6 0.8 1
4 y 3 2 1 30 60 90 120 150 180 210 240 270 300 330 360 390 420 450 480 510 540 x 1 2 3 4