Chapter 4: Trigonometry

§ 1:

Degrees and radians

standard position: vertex=origin, initial side=(positive) x-axis

coterminal angles: same terminal side

measuring size of an angle

one full circle = 360 degrees

one full circle = 2p radians

radian measure = length of arc in circle of radius 1 swept out by the angle

acute, obtuse, reflex angles

A+B = p/2 ; complementary angles (acute)

A+B = p ; supplementary angles (acute,obtuse)

§ 2:

Trigonometric functions

x = cost = cosine of t

y = sint = sine of t

[1/x] = [1/cost] = sect = secant of t [1/y] = [1/sint] = csct = cosecant of t

[y/y] = [sint/cost] = tant = tangent of t [x/y] = [cost/sint] = cott = cotangent of t

Examples:

sin(p/4) = cos(p/4) = Ö2/2

sin(p/6) = 1/2 ; cos(p/6) = Ö3/2

sin(p/3) = Ö3/2 ; cos(p/3) = 1/2

sin(p/2) = 1 ; cos(p/2) = 0 sin(0) = 0 ; cos(0) = 1

Domain of sint, cost : all t

Range: [-1,1]

point on circle corresp. to t+2p is same as point for t

sin(t+2p) = sint ; cos(t+2p) = cost

sint and cost are periodic

symmetry:

cost , sect are even functions

sint, csct, tant, cott are odd functions

x²+y² = 1 (unit corcle): sin² t+cos² t = 1

§ 3:

Right angle trigonometry

sin(q) = a/c = (opposite)/(hypotenuse)

cos(q) = b/c = (adjacent)/(hypotenuse)

tan(q) = a/b = (opposite)/(adjacent)

``SOHCAHTOA''

Copmplementary angle = the `other' angle in a right triangle

sin(p/2-q) = cos(q) , cos(p/2-q) = sin(q)

tan(p/2-q) = cot(q) , cot(p/2-q) = tan(q)

sec(p/2-q) = csc(q) , csc(p/2-q) = sec(q)

( i.e., function(``co-angle'') = ``co-function''(angle) )

§ 4:

Trig functions for any angle

angle q, point (x,y) on terminal side

r = [Ö(x²+y²)]

sin(q) = y/r cos(q) = x/r tan(q) = y/x

reference angle = acute angle that terminal side makes with x-axis

(trig fcn)(q) = (trig fcn)(ref. angle), except possibly for a change in sign:

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III

(x < 0,y > 0) (x > 0,y > 0)

sin(q) > 0 sin(q) > 0

cos(q) < 0 cos(q) > 0

tan(q) < 0 tan(q) > 0

sin(q) < 0 sin(q) < 0

cos(q) < 0 cos(q) > 0

tan(q) > 0 tan(q) < 0

(x < 0,y < 0) (x > 0,y < 0)

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§ 5:

Graphs of sine, cosine

cos(q) = x-value of the points (counter-clockwise) on the unit circle, starting with 1

Graph: note x-intercepts, y-intercept, maximum and minimum; draw a smooth curve

Transformations: y = asin(bx)

vertical stretch by factor of a; amplitude is |a|

amplitude = how far trig function wanders from its `center'

horizontal compression by factor of b; period is 2p/|b|

Translations: just like before

y = cos(x-a) ; translation to right by a

y = cos(x) +a ; translation up by a

§ 6:

Graphs of other trig functions

Transformations (same)

Products: sinx, cosx bounce between -1 and 1; so, for example:

y = xsinx bounces between y = x and y = -x

y = e^-xcosx bounces between y = e^-x and y = -e^-x (`damped' trig function)

§ 7:

Inverse trig functions

f(x) = sinx , -p/2 £ x £ p/2 , is one-to-one

inverse is called arcsinx = angle (between -p/2 and p/2) whose sine is x

sin(arcsinx) = x ; arcsin(sinx) = x if x is between -p/2 and p/2

f(x) = cosx , 0 £ x £ p , is one-to-one

inverse is called arccosx = angle (between 0 and p) whose cosine is x

cos(arccosx) = x ; arccos(cosx) = x if x is between 0 and p

f(x) = tanx , -p/2 < x < p/2 , is one-to-one

inverse is called arctanx = angle (between -p/2 and p/2) whose tangent is x

tan(arctanx) = x ; arctan(tanx) = x if x is between -p/2 and p/2

Graphs: take appropriate piece fo trig function, and flip it across the line y = x

cos(arcsinx) = (cosine of angle whose sine is x) = [Ö(1-x²)] ; etc.

Chapter 5: Analytic trigonometry

§ 1:

Using fundamental identities

Reciprocal: cscx = [1/sinx] secx = [1/cosx] cotx = [1/tanx]

Quotient: tanx = [sinx/cosx] cotx = [cosx/sinx]

Pythagorean: sin² x +cos² x = 1 tan² x +1 = sec² x cot² x+1 = csc² x

Complementarity: sin(p/2 - x) = cos(x) tan(p/2 - x) = cot(x) sec(p/2 - x) = csc(x)

cos(p/2 - x) = sin(x) cot(p/2 - x) = tan(x) csc(p/2 - x) = sec(x)

Symmetry: cos(-x) = cosx sec(-x) = secx

sin(-x) = -sinx csc(-x) = -cscx tan(-x) = -tanx cot(-x) = -cotx

Trig substitution: rewrite expression in x by `pretending' x=trig function

[Ö(a²-x²)] ; write x = asinq, then [Ö(a²-x²)] = acosq

[Ö(a²+x²)] ; write x = atanq, then [Ö(a²+x²)] = asecq

[Ö(x²-a²)] ; write x = asecq, then [Ö(x²-a²)] = ±atanq

§ 2:

Checking trig identities

an equation is solved for the correct values of x

Basic idea: use identities that we already know (like the list above)

convert things to sines and cosines

play with the two sides of the identity

add 0 ! multply and divide by the same expression!

Examples: cscx-sinx = [1/secxtanx]

[(tanx+tany)/(1-tanx tany)] = [(cotx+coty)/(cotxcoty-1)]

§ 3:

Solving trig equations

(single trig function) = (single value)

Wrinkles:

Polynomials: 2cos²x+3cosx+1 = 0 ; (2cosx+1)(cosx+1) = 0

2cosx+1 = 0 or cosx+1 = 0

Trig identities: tanx+secx=4 ; tanx = 4-secx ; square both sides

tan² x (= sec² x-1) = 16-8secx+sec² x = ....

Problem: `ghost solutions' = solutions which `appear' only after manipulating equation

(stupid) Ex: sinx = 1 and (sinx)² = 1 have different sets of solutions!

§ 4:

Angle sum and difference formulas

sin(A-B) = sinAcosB-cosAsinB

cos(A+B) = cosAcosB-sinAsinB

cos(A-B) = cosAcosB+sinAsinB

Note: it is easy to derive any threee formulas from the remaining one, using even/odd and complementarity formulas.

tan(A+B)= [(sin(A+B))/(cos(A+B))] = [(tanA+tanB)/(1-tanAtanB)]

tan(A-B)= [(sin(A-B))/(cos(A-B))] = [(tanA-tanB)/(1+tanAtanB)]

Some uses: complex multiplication! (side trip to part of Section 6.5)

(a+bi)(c+di) = (ac-bd)+(ad+bc)i

pretend z=a+bi=cosA+isinA, z^¢=c+di=cosB+isinB, then this reads

z·z^¢=(cosAcosB-sinAsinB)+(sinAcosB+cosAsinB)i

=cos(A+B) = isin(A+B)

Problem: z=a+bi=cosA+isinA. then a²+b²=sin² A+cos² A=1 (every time)

Solution: think z=a+bi=r(cosA+isinA), where

r²=a²+b²;. i.e, think z« (a,b) (in plane) = point in plane at distance r

from origin, making angle A with (positive) x-axis

i.e., think z=a+bi « (a,b) « (distance,angle) ; polar coordinates

then complex multiplication multiplies distance and adds angles:

(r(cosA+isinA))(r^¢(cosB+isinB)) = (rr^¢)(cos(A+B)+isin(A+B))

Another use: find values of trig functions at new angles:

Example: 105^° = 60^° +45^° (i.e. 7p/12 = p/3+p/4), so

cos(7p/12) = cos(p/3+p/4) = cos(p/3)cos(p/4)-sin(p/3)sin(p/4) =

(1/2)(Ö2/2)-(Ö3/2)(Ö2/2) = (Ö2-Ö6)/4

§ 5:

Multiple angle, product-to-sum formulas

sin(2A) = sin(A+A) = 2sinAcosA

cos(2A) = cos(A+A) = cos² A-sin² a = 2cos² a-1 = 1-2sin² A

Triple angle? sin(3A)=sin(2A+A)=....

sin² x=(1-cos(2x))/2 , cos² x=(1+cos(2x))/2 ; these give

Half-angle formulas:

sin(x/2) = Ö{(1-cosx)/2} ; cos(x/2) = Ö{(1+cosx)/2}

tan(x/2) = [sinx/(1+cosx)] = [(1-cosx)/sinx]

Product-to-sum formulas:

sin(A+B)+sin(A-B) = 2sinAcosB, so

sinAcosB= [1/2](sin(A+B)+sin(A-B)) Simlarly,

cosAcosB = [1/2](cos(A+B)+cos(A-B)), and

sinAsinB = [1/2](cos(A-B)-cos(A+B))

Sum-to-product formulas:

set A+B = x, A-B = y (solve: A=[(x+y)/2], B=[(x-y)/2]), plug in above!

sinx+siny = 2sin[(x+y)/2]cos[(x-y)/2]

cosx+cosy = 2cos[(x+y)/2]cos[(x-y)/2]

cosx-cosy = 2sin[(x+y)/2]sin[(x-y)/2]

OK, so what's the point? It's alot easier to remember what these formulas (in the previous two sections) say if you remember where they come from. We built all of these formulas up from one formula; cos(A-B) = ..... . If you remember how each follows one from the other, then you don't `have to' remember the formula!