The Unit Circle
Right Triangle Trigonometry Hypotenuse Opposite Side Adjacent Side S O H C A H T O A i p y o d y a p d n p p s j p n p j sin opposite side hypotenuse csc hypotenuse opposite side cos adjacent side hypotenuse sec hypotenuse adjacent side tan opposite side adjacent side cot adjacent side opposite side
Right Triangle Trigonometry xy, r y x cos x y r r x y adjacent hypotenuse x r sec Reciprocal Relations hypotenuse r adjacent x cos sin opposite hypotenuse y r hypotenuse r csc opposite y sin tan opposite adjacent y x adjacent x cot opposite y tan cos sin Pythagorean Identities tan sec cot csc
Unit Circle / Triangle Trigonometry (, ) xy = (cos θ,sin θ) y θ x adjacent side x cos θ= = = x hypotenuse hypotenuse sec θ= = adjacent side x opposite side y sin θ= = = y hypotenuse hypotenuse csc θ= = opposite side y tan opposite side θ= = adjacent side y x adjacent side cot θ= = x opposite side y By the Pythagorean Theorem: x + y = ( θ) ( θ) cos + sin = cos θ + sin θ =
Def inition of the Six Trigonometric Functions Riglrt trianglectefnition.r, *'here 0 < 0 < rr/. sing=-g nyp. opposite adi. cos 0 = lryp. tan0: ODD, -taoj, Circular function definitions, where 0 is any angle. Reciprocaldentities. slnu: csc u csc u ll sln u VX tan0: cot0:- xy secu:- cos u YT sin0= csc0:- ry XT cos0:- scc0=rx sec u I sinusinv:rlcos(u-v)- I tanu:- cot u I cotu:- tun u Tangent and Cotangent ldentities sin u tatru:- cos u cotu cos u stn u Pythagorean ldentities sinzu*cosu:l I -f tan u : sec u Cofunction ldentities. lrr \ ttn\z - u) : cos u lrr \.t.\t -,,) : sec rl lrr \ t..\t - u) - csc u I * cotz u = cscz u cost; lr - u \ :sinu \r, /, lrr \ ttn(t - ui = cot tl / ln \ cot\z - u :tanu ) Negative Angle ldentities sin(-u) : -sin u cos(-u) = cos u csc(-u) : -csc u tan(-u) = -tan u sec(-u) = sec u cot(-u) = -cot u Surn and Difference Formulas sin(utv) : sin u cos v -t cos u sin v cos(utv) : cos u cos v T sin u sin v tall(uiv) : Arljucerrt tanu+tunv ;-=-_ lttanutanv csc6: sec0= cot 0 : Iryp. opp. hvg adj, adj. opp. r Jl, Jit \-'ftf) r,ft l\ \--Ttl) (-,0) (0, ) rl v5 \r / rrt Jlt \-f,t) rrt r 600 T \T,i) 45' 300 330o -E-,r 5' 350 r rt lr L4 z4o" rrt rtt e+,-*) \Tt-TJ --- /-r -f\ rr - /5r \r \ ), zt (0,_l) \ ' / (+,-L) Double-Angle Formulas sinu:sinucosu cos u : cos u - sin u: cos u - t - l'- sin u tanu tirnu:.-- I - tilrt'u Power-Reducing Formu las. l-cosu sln'u -- *cos u cos. " u = --;- ^ L I -cos u tan" u : T+;;t Su m-to-product Forrnulas sin u t sin v = ',"(+)."r(+) \4/ \/ sin u - sin y =."'(+)''"(?) cos u -r- cos v = z'"'(+)."'(+) \z/ \ L/ cos u - cos v: -zt'"(+) t,.(+) \ Z/ \ L / Product-to-Sum Formulas cos u cos v = jt.ort,r-v) * sinucosv:jtrintu*v)t cos u sin v -- jtr,ntu*u) - cos(u -F v)l cos(u*v)l sin(u - v)l sin(u - v)l (,0)
Circle: Trigonometric Identities x y r Reciprocal Relations Pythagorean Identities sin y opp r hyp cos x adj r hyp tan y opp x adj sin cos Opposite Angle Relations csc sec cot sin cos tan csc x sin x sec x cos x cot x tan x cos sin tan sec Double Angle Relations Even Relations cos( ) cos sin sin cos sec( ) sec Odd Relations sin( ) sin Sum Relations tan( ) tan cot( ) cot csc( ) csc Difference Relations cot csc sin( A B) sin Acos B cos Asin B sin( A B) sin Acos B cos Asin B cos( A B) cos Acos B sin Asin B cos( A B) cos Acos B sin Asin B tan( A B) tan A tan B tan Atan B tan( A B) cos cos sin cos sin tan tan tan tan A tan B tan Atan B Reducing Powers sin cos cos cos tan cos cos Product-to-Sum Relations cos A cos B cos( A B) cos( A B) sin A sin B cos( A B) cos( A B) sin A cos B sin( A B) sin( A B) cos A sin B sin( A B) sin( A B) Half Angle Relations cos sin cos cos cos sin cos tan sin cos cos Sum-to-Product Relations A B A B sin A sin B sin cos A B A B sin A sin B cos sin A B A B cos A cos B cos cos A B A B cos A cos B sin sin
Table of Values for cos θ, sin θ and tanθ Degree Radian θ θ cosθ sinθ tanθ 0 0 0 0 30 3 6 3 3 45 4 60 3 3 3 90 0 is undefined 0
REFERENCE ANGLES fi,e*ru {..' Jo crcrf alrelwrgl, p i$ t*. rcdmcn* rrg}c for 0. Belol+ is a chart that will help in the easy calculation of reference angles. For angles in the first quadrant, the reference angle pis equal to the given angle 8. For angles in other quadrants, reference angles are calculated this way: q{ed?rnt I I ill tv p (re,ference ffigle) F*o F = tlo'8 F* 8'rac F" ltc'o FigurA 4.A: Ho* ta calculate thz refercnce angle pfar any angle I detweea Q and n radians. http:/hwvw.sparknotes.com/math/trieonometr.v/trigonometricfunctions/section4.rhtml
The Unit Circle cos( θ) and sin( θ ) Calculator Each Angle Increases by 0
MAC4 Sample Test Part page Name Fill in the following table with the appropriate values on a Unit Circle. No Calculators Allowed. θ in degrees θ in radians Reference Angle (in degrees) Coordinates (x, y) sin θ cos θ tan θ ) 60 XXX XXX XXX ) 7 6 XXX XXX XXX 3), XXX XXX XXX 4) 0 5) 4 6) XXX ( 0, ) 7) sin θ= θ in QII 8) 9) 3 cos θ= θ in QIV tan θ = θ in QII 0) 40 ) ) 5 3 sin θ= θ in QIV
page MAC4 Sample Test Part Name 3) A) Convert 00 to radian measure. B) Convert 0 to degree measure. 4) Convert 57 4 5 to decimal degrees. Round to 3 decimal places. 5) Convert 3.74 to Degrees/Minutes/Seconds 6) The terminal side of θ in standard position passes through the point (, 5). A) Find the values of the six trigonometric functions of θ. B) Find the value of θ rounded to decimal places. 7) Use your calculator to find the values of the following. Round answers to decimal places. A) sin(36.4 ) B) cos(5 3 ) C) sec(68 ) D) cot(80 )
MAC4 Sample Test Part 8) Find the following trig ratios, based on the triangle to the right. A) cos A = B page 3 B) tan B = 3 5 C) sin A = A C D) sec B = Solve the following right triangles B A = a = 9) B = b = 5 3 C = c = A 4 C A = a = B 0) B = b = 7 C = c = A 0 C
MAC4 Sample Test Part ) A six foot tall man casts a 5 foot long shadow. A tree casts a 00 foot long shadow. How tall is the tree? page 4 ) A hiker leaves camp, looks on his compass, and walks on a bearing of 300. If the dot represents camp, draw a ray showing the direction the hiker starts walking. N W E S 3) Find the height of the building 30 00 feet h
page 5 The following Unit Circle is given for your use and reference only. No points are given for values filled out on the circle.
Transformations of y = f(x) = sin x Parent Function Vertical Stretch Vertical Compression y = f(x) = sin x y = f(x) = sin x y = f(x) = sin x Horizontal Shift Right Horizontal Shift Left 4 Horizontal Shift Right y = f(x ) = sin(x ) 4 4 y = f(x + ) = sin(x + ) y = f(x ) = sin(x ) Vertical Shift Up Vertical Shift Down Reflection about the x-axis y = f(x) + = + sin x y = f(x) = + sin x y = f(x) = sin x Horizontal Compression Horizontal Compression Horizontal Stretch y = f(x) = sinx y = f(4x) = sin4x y = f( x) = sin x
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Graph: y 3sin x Domain: Range: Amplitude: Period:
Transformations of Sine and Cosine Find two equations for each graph. Use SINE for the st equation, then use COSINE for the nd equation. y y y y y y y y y y y y y y y y y y
MAC4 Test Sample Part page Name Fill in the following table with the appropriate values on a Unit Circle. No Calculators Allowed. θ in degrees ) 0 ) θ in radians 4 Reference Angle (in degrees) 3) XXX ( 0, ) 4) 5) 6) Coordinates (x, y) sin θ cos θ tan θ sin θ= θ in QII 3 cos θ= θ in QIV tan θ = θ in QII 7) A) Convert 00 to radian measure. B) Convert 0 to degree measure. 8) Formulas: Arc Length: Area of a Sector: Linear Velocity: Angular Velocity: s = A = V = = = w = ω = 9) Graph y = sin( x) + 0) Graph y = cos( x)
MAC4 Test Sample Part Calculator is allowed / required on Part page Name ) Graph y = sin x + State the Domain State the Range State the Amplitude State the Period 4 ) Graph y = 3cos ( x) State the Domain State the Range State the Amplitude State the Period Show all asymptotes with dotted lines. 3) Graph y = cot x State the Domain State the Range State the Period
Show all asymptotes with dotted lines page 3 4) Graph y = tan( x) State the Domain State the Range State the Period 5) Graph y ( x ) = sin + 4 State the Domain State the Range State the Amplitude State the Period 6) Graph y ( x ) = 3cos ( ) + 4 State the Domain State the Range State the Amplitude State the Period
Show all asymptotes with dotted lines page 4 7) Graph y = sec( x) State the Domain State the Range State the Period 8) Graph y = csc ( x) State the Domain State the Range State the Amplitude State the Period 9) Graph y = 3sec( x) State the Domain State the Range State the Amplitude State the Period
Formulas: s = rθ A = r θ r t = d r = d s rθ v = = t t t = rω ω= θ t page 5 0) Find the length of the arc of a circle of diameter 0 feet, intercepted by a central angle of 50. A) Exact Length B) Length rounded to decimal places. ) A young boy cuts a piece of apple pie (a sector). The diameter of the circular pie pan is 9 inches. He noticed that the angle of the pie cut in the center was exactly 60. Find the area of the boy s piece of pie. ) Suppose that a jogger is running around a circular track, 50 m in radius. The jogger runs 3 of the way around the track in 60 seconds, then stops gasping for air. A) How fast was the jogger running (in meters per second m/s)? B) A coach standing in the center of the track area was videotaping the runner. How fast was the coach spinning (in radians per second)?
page 6 The following Unit Circle is given for your use and reference only. No points are given for values filled out on the circle.