Trigonometry Revision Sheet Q of Pper The Bsis - The Trigonometry setion is ll out tringles. We will normlly e given some of the sides or ngles of tringle nd we use formule nd rules to find the others. Some rules, suh s the Tn, Cos nd Sin rtios hve to e lerned while others suh s the Sine Rule nd Cosine Rule n e found in the tles. Pge 9 of the tles ontins mny of the formule you need, so use this pge when prtiing questions. It is very importnt to prtie questions to eome fmilir with when to use eh rule. There re silly types of tringle we n e sked to solve.. Right ngled tringles using the Tn, Cos, Sin rtios. Non-right ngled tringles using Sine rule.. Doule tringles using omintion of the ove.. Ares of tringles nd setors using formul in the tles. Things to rememer: Angles of stright-line sum to 80 Angles of tringle sum to 80 0 0 so in this emple = 80 = so in this emple = 80 (0 + 0) = 80 A right ngle tringle is tringle where one side is perpendiulr (t 90 ) to nother. Angle of depression Diretions N W W0N NE E S0W Angle of elevtion S
In ll of the trigonometry questions the first step is lwys to drw rough sketh of the tringle nd put in n X on the side or ngle you re looking to find. Pythgors - In right ngle tringle the squre of the hypotenuse is equl to the sum of the squres of the other two sides. Bsilly the long side squred is equl to the squres of the other two sides dded together. A Adjent The hypotenuse is lwys the longest side of right-ngled tringle. The opposite side is found y drwing n rrow through the ngle. The djent is the remining side. If we re given two sides of right-ngled tringle we n use Pythgors to find the remining side. When nswering ny trigonometry question lwys your rough sketh nd mrk in the hypotenuse, the opposite nd djent sides. Find the vlue of in the elow emples. Emple Emple 9 9 9
The Tn, Sin nd Cos Rtios A Adjent TnA Adjent SinA CosA Adjent Old Toms Aunt On St Her And Cot Ht These rtios re not in the tles nd must e lerned off y hert. If we hve one rtio (s frtion) we n use it to find the other two. Alwys drw rough sketh to help you work out wht rtio to use. Emple Find Tn A, Cos A nd Sin A of the following tringle. A TnA Adjent SinA CosA Adjent TnA SinA CosA. Lel the sides opposite, djent nd hypotenuse. Write down the rtios.. Insert the relevnt numers. Sometimes ll the sides will not e given to you nd you must work them out using Pythgors. Emple CosA Find TnA nd SinA Hyp. A Adj. Opp. CosA Adj. CosA Hyp. Adjent = = so we drw rough sketh using s the djent side nd s the hypotenuse. We now use Pythgors to find the missing side, X. 9 9 = TnA Adjent TnA SinA SinA
Clultor Work You need to e very omfortle using your lultor wy efore the em s they differ gretly from model to model. Alwys mke sure your lultor is in DEG mode. There re two min things we need to e le to do: Find Tn, Sin or Cos of given ngle. Emple Find Cos. Press Cos. then type. then = = 0.08 (round off to deimls) Emple Find Sin '. Press Sin. then type. then the DMS utton. then. then = Find the ngle given the Tn, Sin or Cos. Emple Cos A = 0.707 Find A A =. Press nd Funtion. Press Cos. then type 0.707. then = If you get deiml nswer you n hnge it into degrees nd minutes y gin pressing the DMS utton OR the nd Funtion then the DMS utton. = 0.8 (round off to deimls)
Solving right-ngled tringles - We n use our rtios Tn, Sin nd Cos to find missing sides or ngles of right-ngled tringles.. To do this we first drw rough sketh of the tringle filing in ny informtion we know.. We put n X net to the ngle or side we re looking for.. We deide wht rtio Tn, Sin or Cos is relevnt.. We solve for. Emple is right-ngled tringle with = m nd 0. Find. We first drw rough sketh lelling the sides nd pling n X on the side we re looking for. The opposite () nd hypotenuse (X) re the sides involved so we use the Sin rtio. X 0 SinA Write down the formul Sin0 Insert the known vlues Sin0. 9m 0.80 Cross multiply The vlue of Emple pqr is right-ngled tringle with pr = 8 nd pq = 0. Clulte the vlue of pqr. q We first drw rough sketh lelling the sides nd this time pling n X in the ndle we re looking for. The djent (8) nd hypotenuse (0) re the sides involved so we use the Cos rtio. r 8 0 X p Adjent CosA Write down the formul 8 CosX Insert the known vlues 0 CosX 0.8000 Chnge frtion into deiml X = nd Funtion Cos 0.8000 Use lultor to find Angle X = ' The ngle pqr
Angles of 0,, 0 nd 90. Some questions require nswers to e left in frtion form without the use of lultor. Any time you see the Sin, Cos or Tn of the ngles of 0,, 0 or 90 you n look up their frtion/ surd vlues in the tles nd insert them into the question. In the tles on pge 9, stnds for 80. A 0 Cos A - 0 Sin A 0 0 Tn A 0 0 Emple Find in frtion form. sin os os 0. tn 0 sin 0. sin os os 0. tn 0 sin 0. ( ) ( ) ( )... () ()
Angles greter thn 90 - The Unit Cirle For questions involving ngles greter thn 90 we use the unit irle. The letter in eh qudrnt determines whether Sin, Cos or Tn is negtive or positive in tht qudrnt. Between 0 nd 90 All re positive Between 90 nd 80 Sin is positive, Tn, Cos negtive Between 80 nd 70 Tn is positive, Sin, Cos negtive Between 70 nd 0 Cos is positive, Sin, Tn negtive To find the rtio of n ngle greter thn 90 we use the following steps.. Mke rough sketh of the ngle in the unit irle.. Use to find if the rtio is positive or negtive.. Find the referene ngle (the ngle etween your skethed line nd the is).. Use lultor to find required rtio of this referene ngle nd use the sign from step. Emple Find the vlue of Sin 0. Drw rough sketh of 0. In the nd qudrnt Sin is positive. The referene ngle is ( 80 0) 0 Use your tles to find the Sin 0 Use the positive sign from ove To find the ngles (there will e two) given the rtio we use the following steps.. Ignore the sign nd use your lultor to find the referene ngle.. Look t the sign nd use to see wht qudrnts your two ngles lie in.. Drw the two ngles they will e the referene ngle distne from the is. Emple Find the two vlues of A, given tht Cos A = -0.80 nd 0 A 0 Ignoring the sign use your lultor to find the referene ngle using nd Funtion Cos 0.80 = 0-0.80 is negtive nd Cos is negtive in the nd nd rd qudrnts. Drw the two ngles: In the nd qudrnt 80 0 0 In the nd qudrnt 80 0 0 7
Are of tringle The re of tringle n e found using the following formul (Pge 9 of the tles) SinC where nd re the sides of tringle nd C is the ngle etween them. Emple Find the Are of tringle yz C y 9 8 z SinC Write down formul (9)(8) Sin Insert vlues (9)(8)(.7880) Work out Sin 8.7m Are of the tringle If we re given the re we n use the formul to find side or ngle. Emple The Are of tringle pqr is m. Find the length of the side pq q 0 p r SinC Write down formul equl to ( )(0) Sin Insert vlues inluding ( )(0)(.7) Work out Sin.87 Simplify 8.7m.87 Divide ross y to find the side Emple The Are of tringle pqr is 0m. Find the ngle of qpr p X r q SinC 0 Write down formul equl to 0 ()() SinX 0 Su in known vlue 78SinX 0 Simplify 0 SinX 78 Divide ross y 78 SinX 0.0 0 divided y 78 = 0.0 X = nd Funtion Sin 0.0 = 9 ' The ngle qpr 8
Sine Rule SinA B Tles Pge 9 SinB C A We use the Sine formul to find missing side or ngle when we hve een given sides nd one of the opposite ngles OR two ngles nd one of the opposite sides of tringle. The formul silly sys tht Any side = Any other side The Sine of the opposite ngle The Sine of its opposite ngle Emple In the tringle pqr, pq = m, pqr 0 nd qrp. Find pr. SinA SinB Write down formul Sin Sin0 Insert known vlues sin sin 0 Cross Multiply (0.7) = (0.80) Clultor for Sine s (0.80) 0.9 0.7 0.7 Divide ross y 0.7 = 8. Emple In the tringle pqr, pq = m, pr = m nd pr = 8.m pqr 7. Find the prq SinA SinB Write down formul Sin7 SinX Insert known vlues sin X sin 7 Cross Multiply sin 7 SinX Divide ross y (0.99).9 SinX = 0.879 Simplify Sin X = 0.879 X = nd Funtion Sin 0.879 =.88 Clultor X = ' Vlue of prq 9
Are of Setor - A setor is prt or slie of irle. To find the re of setor we tully use informtion from the Are nd Volume question. Rememer tht the re of irle is r To find the setor (whih is only frtion of the irle) we multiply r y the frtion. Therefore Are of Setor = where is the ngle of the setor 0 r Emple Find the re of the setor o where o is the entre of irle with rdius 7. Use =. Write down the formul 0 r o 90 7 90 Su in known vlues (.)(7) 0 0. (.)(9) Simplify Are of the setor 8.m Some questions involve using the re of setor formul long with the re of tringle formul. Emple Find the shded re in the irle with entre o nd rdius 7. Use 7 o 7 0 7 Are of shded region = Are of setor Are of tringle Are of setor = r 0 0 = (7) = (7)(7) Sin 0 0 7 0 = (7) = (7)(7)(0.87) 0 7 = 0.7 =0.(7)(7)(0.87) =.8m =.m Are of shded region = Are of setor Are of tringle Are of tringle = SinC =.8m -.m =.8 m 0
Doule tringles The rules for solving doule tringles re the sme s those in the previous setions. Mostly they will e two-prt questions, one prt sking you to solve smll tringle within the lrge one using the osine or sine rule nd the other prt sking you to find n ngle or side on the lrger tringle. Emple In the tringle, d =, =, (i) Find d (ii) Find d 70 nd d 70 m Notie tht this is wht we men y doule tringle. Two smller tringles d nd d mke up the lrger tringle. m d 70 m m d d The first thing we do is fill in ny missing informtion we n knowing tht stright lines dd up to 80 nd the three ngles in tringle dd to 80. m X m m d d 70 70 X 8m m m If were sked we ould lso find the re of the tringles or d using the formul. 80 (70 + ) = ( ngles dd to 80) Fill in X for the missing vlue d Now we n use Sine Rule with the tringle d SinA SinB Sin Sin70 (Sin) = (Sin70) (0.707) = (.997) (.997) = 8m d = 8m (i) 0.707 If we put 8m in for d we n see tht the totl length of is m. Now we n use the Cosine rule with tringle CosA ()() Cos0 ()()(0.) 7 08 = 0.m =0.m (ii)