**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

x^{2}+y^{2} = 1 (unit corcle): sin^{2} t+cos^{2} 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(x^{2}+y^{2})]

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:

height1.8in width.4pt depth0pt

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

height.4pt width4.5in depth0pt

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)

IIIIV

- § 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^{-x}cosx 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-x^{2})] ; 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: sin^{2} x +cos^{2} x = 1 tan^{2} x +1 = sec^{2} x
cot^{2} x+1 = csc^{2} 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(a^{2}-x^{2})] ; write x = asinq, then [Ö(a^{2}-x^{2})] = acosq

[Ö(a^{2}+x^{2})] ; write x = atanq, then [Ö(a^{2}+x^{2})] = asecq

[Ö(x^{2}-a^{2})] ; write x = asecq, then [Ö(x^{2}-a^{2})] = ±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: 2cos^{2}x+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

tan^{2} x (= sec^{2} x-1) = 16-8secx+sec^{2} 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 a^{2}+b^{2}=sin^{2} A+cos^{2} A=1 (every time)

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

r^{2}=a^{2}+b^{2};. 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) = cos^{2} A-sin^{2} a = 2cos^{2} a-1 = 1-2sin^{2} A

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

sin^{2} x=(1-cos(2x))/2 , cos^{2} 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!

File translated from T