Higher Maths EF1.2 and RC1.2 Trigonometry - Revision

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Bennett Scot White
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1 Higher Maths EF and R Trignmetry - Revisin This revisin pack cvers the skills at Unit Assessment and exam level fr Trignmetry s yu can evaluate yur learning f this utcme. It is imprtant that yu prepare fr Unit Assessments but yu shuld als remember that the final exam is cnsiderably mre challenging, thus practice f exam cntent thrughut the curse is essential fr success. The SQA des nt currently allw fr the creatin f practice assessments that mirrr the real assessments s yu shuld make sure yur knwledge cvers the sub skills listed belw in rder t achieve success in assessments as these revisin packs may nt cver every pssible questin that culd arise in an assessment. Tpic Unit Sub skills Questins Applicatin f: The additin r duble angle frmulae Trignmetry: Manipulating trignmetric expressins EF Applicatin f: Trignmetric identities nvert nvert acs x bsin x t k cs( x ) r sin( x ) in st quadrant k, a cs x b sin x t k cs( x ) r sin( x ) in any quadrant k, k 0 k , - Trignmetry: Slving trignmetric expressins R Slve trignmetric equatins in degrees, invlving trignmetric frmulae, in a given interval Slving trignmetric equatins in degrees r radians, including thse invlving the wave functin r trignmetric frmulae r identities, in a given interval 9, 0,, When attempting a questin, this key will give yu additinal imprtant infrmatin. Key Nte Questin is at unit assessment level, a similar questin culd appear in a unit assessment r an exam. Questin is at exam level, a questin f similar difficulty will nly appear in an exam. * The questin includes a reasning element and typically makes a questin mre challenging. Bth the Unit Assessment and exam will have reasning questins. If a star is placed beside ne f the abve symbls that indicates the questin invlves sub skills frm previusly learned tpics. If yu struggle with this questin yu shuld g back and review that tpic, reference t the tpic will be in the marking scheme. Questin shuld be cmpleted withut a calculatr. Questin shuld be cmpleted with a calculatr. Questins in this pack will be rdered by sub skill and typically will start ff easier within that subskill and then get mre challenging. Sme questins may als cver several sub skills frm this utcme r even include sub skills frm previusly learned tpics (dented with a *). Questins are gathered frm multiple surces including nes we have created and frm past papers. Extra challenge questins are fr extensin and are nt essential fr either Unit Assessment r exam preparatin. JGHS H EF and R Revisin

2 FORMULAE LIST ircle: The equatin y gx fy c 0 g f c g, f x represents a circle centre The equatin x a y b r represents a circle centre a, b and radius r. and radius Scalar Prduct: a.b a b cs, where is the angle between a and b r a.b ab ab ab where a a a a and b b b b. Trignmetric frmulae: sin A B csa B sin Acs B cs Asin B cs Acs B sin Asin B sin A sin Acs A cs A cs A sin A cs sin A A Table f standard derivatives: f x f x sin ax cs ax acs ax asin ax Table f standard integrals: x f f xdx sin ax cs ax cs ax a sin a ax JGHS H EF and R Revisin

3 Q Questins Marks Using the right angled triangles belw, (a) Evaluate sin( c d) (b) Evaluate cs( c d) (c) Evaluate tan( c d) Fr the triangles ppsite, find the exact value f sin( x y). The gradient f OA is The gradient f OB is Prve that cs AOB. Fr the triangle ppsite, find the exact value f sin x Given that tan x, find the exact value f cs x JGHS H EF and R Revisin

4 6 Prve the fllwing identities (a) sin cs sin (b) cs sin sin (c) cs sin cs sin cs (d) sin sin cs 7 Prve the fllwing identities (a) cs cs cs (b) sin tan cs (c) sin x cs xsin x cs x sin x Express Express Express Express Express Express sin x cs x in the frm k sin( x a), where k 0 and sin x cs x in the frm k cs( x a), where k 0 and 0 x 60 0 sin x cs x in the frm k sin( x a), where k 0 and 0 x sin x cs x in the frm k sin( x a), where k 0 and 0 x 60 x 60 cs x sin x in the frm k cs( x a), where k 0 and 0 x cs x sin x in the frm k sin( x a), where k 0 and 0 x Slve cs x 0 fr 0 x 80 JGHS H EF and R Revisin

5 Slve sin x 0 fr Slve tan x fr 0 0 x 60 x 60 Slve sin x cs0 cs xsin 0 fr Slve sin x cs x 0 fr Slve cs x cs x fr 0 0 Slve sin x 0 0 x Slve cs x 0 fr 0 x (a) Shw that cs x sin x x 60 x 60 0 x 60 can be written in the frm x 6 6 (b) Hence slve cs x sin x 0 x 60 cs JGHS H EF and R Revisin

6 (a) The graphs y f (x) and y g(x) are shwn belw. y f (x) is f the frm f ( x) a bsin x. State the values f a and b. (b) g( x) cs x. Find the crdinates f intersectin f y f (x) and y g(x) fr 0 x 9 [END OF REVISION QUESTIONS] [G t next page fr the Marking Scheme] JGHS H EF and R Revisin

7 Where suitable, yu shuld always fllw thrugh an errr as yu may still gain partial credit. If yu are unsure hw t d this ask yur teacher. Q Marking Scheme (a) Find missing sides and Expand sin( c d) Substitutin Slutin (b) Expand cs( c d) (c) 8 Knw hw t find tan 8 9 Substitute and evaluate 9 Ntes: sin c cs d cs csin d 6 6 cs c cs d sin csin d 6 6 sin( c d) tan( c d) cs( c d) Find missing sides and 9 Expand sin( x y) sin x cs y cs xsin y Substitutin Slutin Ntes: 9 JGHS H EF and R Revisin

8 Interpret length OA and OB Interpret and expand cs AOB and cs AOB cs cs cs sin sin Substitutin Slutin Ntes: Find the missing side 0 Expand sin x Substitute Slutin Ntes: Suitable strategy (eg triangle) sin x cs x x Any expansin f cs x Eg cs x sin x substitutin evaluate Ntes: JGHS H EF and R Revisin

9 6 (a) Expand left hand side sin sin cs cs Knw that sin cs sin cs Knw that sin cs sin and cmplete prf sin RHS sin cs sin Therefre (b) Expand left hand side sin sin cs cs Knw that sin cs sin cs 6 Knw that sin cs sin and cmplete prf 6 sin RHS cs sin sin Therefre (c) 7 Expand left hand side 7 cs cs sin cs sin sin 8 Simplify 8 cs sin 9 Knw that cs sin cs and cmplete prf 9 cs RHS (d) 0 Expand brackets 0 sin Knw t use cs Therefre cs sin cs sin cs sin cs mplete prf Ntes: cs RHS Therefre sin sin cs JGHS H EF and R Revisin

10 7 8 (a) Expand left hand side cs Knw t write in frm t use cs cs cmplete prf (b) Expand left hand side Knw that sin cs 6 Knw that sin cs sin and cmplete prf cs 6 cs RHS Therefre cs cs cs sin cs cs sin sin cs cs sin cs sin sin cs sin tan RHS cs cs sin Therefre tan cs (c) 7 Expand left hand side 7 sin 6 sin cs cs 8 Knw that sin cs 8 6 sin cs 9 Knw that sin cs sin and cmplete prf Ntes: 9 sin RHS Therefre sin x cs x sin x cs x sin x Expand k sin( x a) k sin x cs a k cs xsin a stated mpare cefficients k cs a and k sin a stated Prcess fr k k 9 Prcess fr a and state answer a 68 therefre sin x cs x 9 sin x 68 Ntes: JGHS H EF and R Revisin

11 9 0 Expand k cs( x a) k cs x cs a k sin xsin a stated mpare cefficients Prcess fr k Prcess fr a Ntes: and state answer k cs a k a 60 and k sin a stated therefre sin x cs x cs x 60 Expand k sin( x a) k sin x cs a k cs xsin a stated mpare cefficients Prcess fr k k cs a k 7 and k sin a stated Prcess fr a and state answer a therefre sin x cs x 7 sin x Ntes: Expand k sin( x a) k sin x cs a k cs xsin a stated mpare cefficients k cs a and k sin a stated Prcess fr k Prcess fr a Ntes: k and state answer a 00 therefre sin x cs x sin x 00 Expand k cs( x a) k cs x cs a k sin xsin a stated mpare cefficients k cs a and k sin a stated Prcess fr k k Prcess fr a and state answer a 6 therefre cs x sin x cs x 6 Ntes: JGHS H EF and R Revisin

12 6 7 Expand k sin( x a) k sin x cs a k cs xsin a stated mpare cefficients k cs a and k sin a stated Prcess fr k Prcess fr a Ntes: and state answer Start t slve and find an angle Slve fr Slve fr x Ntes: x Start t slve and find an angle Find the first tw angles fr Find tw further angles fr Slve fr x Ntes: x x Start t slve k 8 a therefre cs x sin x 8 sin x sin x and sin x, x, 67 sin x and sin x 0, 0 x 0, 0, 90, 0 x, 7, 9, tan x and Find tw angles x 0, 0 Find ther tw angles x, 0, 0 Ntes: Start t simplify sin x 0 Find first angle x 0 8 Find secnd angle x 0 8 Slve x 8, 8 Ntes: tan x JGHS H EF and R Revisin

13 8 9 0 Expand sin x sin xcs x cs x 0 cs x 6sin x factrise 0 Slve fr ne factr Slve fr secnd factr Ntes: Substitute csx cs x 0 x 90, 70 sin x 6 x 6, 0 6 cs x cs x Write in standard frm 6 cs x cs x 0 Factrise 6 cs x cs x 0 Slve fr ne factr Slve fr secnd factr Ntes: Start t slve Find tw angles Find ther tw angles Ntes: Start t slve and find an angle cs x x 0, 60 cs x 6 x 6 sin x and x, x,, 6 6, 6 sin x cs x and cs Find the first tw angles fr x x 9, 7, Find tw further angles fr x x 9, 7, 8 9, Slve fr x Ntes: x 0 96, 9, 09, 7 JGHS H EF and R Revisin

14 (a) Expand k cs( x a) k cs x cs a k sin xsin a stated mpare cefficients k cs a and Prcess fr k Prcess fr a and cmplete prf (b) Link equatin in part (a) t part (b) k a 6 6 therefre k sin a stated (evidence f wrking required) (evidence required) cs x sin x cs x 6 6 cs x sin x cs x Find first angle 6 x Find secnd angle 7 x 6 6 6, slve 8 x 90, Ntes: (a) State a State b (b) Equate graphs and write in standard frm Valid strategy fr simplifying RHS a b Eg cs x sin x cs x sin x cs x sin x k cs( x a) k cs x cs a k sin xsin a mpare cefficients k cs a and k sin a stated 6 Prcess k 6 k 0 7 Prcess a 7 a and 8 Start t slve 8 0 cs x and cs 0 9 Find angles 9 x 6, 0 0 Find x crdinates 0 x, 8 Ntes: Find y crdinates by substituting int either y f (x) r y g(x), 09 and 8, 0 [END OF MARKING SHEME] JGHS H EF and R Revisin

MATH PRACTICE EXAM 2 (Sections 2.6, , )

MATH PRACTICE EXAM 2 (Sections 2.6, , ) MATH 1050-90 PRACTICE EXAM 2 (Sectins 2.6, 3.1-3.5, 7.1-7.6) The purpse f the practice exam is t give yu an idea f the fllwing: length f exam difficulty level f prblems Yur actual exam will have different