Trigonometry Flashcards
Sine rule to find an angle
SinA over a is equal to SinB over b
Sine Rule to find a side
a over sinA is equal to b over sinB
Cosine rule to find a side
one buddy buddy
a squared = b squared plus c squared minus two bc cosA
Complementary angles
Sin♎️= cos(90-♎️) and visa versa Tan♎️= cot(90-♎️) and visa versa Cosec♎️= sec(90-♎️) and visa versa
Types of trigonometric equations
1) single angle
2) multiple angle
3) square angle
4) compound angles
Single angles
Solve for ♎️ within a given domain.
Multiple angle
When solving a multiple angle you must
1) multiply the domain by the coefficient of ♎️ and therefore introduce a new pronumeral
e. g. Sin2♎️=1/2 for 0 sinx=1/2 for 0<720
2) solve for the new pronumeral e.g. x
3) substitute the unknown term back in for x and solve
Square angle
When solving a square angle you must move the square outside the brackets and root the other side. You will end up with a plus minus, ie both positive and negative angles in all fourvquadrants
Compound angle
In a compound angle a term inside brackets will see ♎️ added, subtracted, multiplied, divided by numbers. Much like double angles, whatever is happening to ♎️ must happen to the domain when the bracketed term is let to be x or a pronumeral.
Trig graphs sinx and cosecx
Sinx- walks up the mountain and down the valley
Cosecx- suction cups
Trig graphs cosx and secx
Cosx- gets cut short at 1/2 a mountain ‘late bloomer’
Secx- 1/2 suction cups
Trig graphs tanx and cotx
Tanx- 3 strip vertical flag
Cotx- rebel goes opposite way
Trig identity 1
Tan♎️=Sin♎️/cos♎️
Trig identity 2
Cot♎️=Cos♎️/sin♎️
Trig identity 3
Sin squared ♎️ + cos squared ♎️ = 1
Trig identity 4
1 + cot squared ♎️= cosec squared ♎️
Trig identity 5
Tan squared ♎️ + 1= sec squared ♎️
Sin sum of angles
Sin(x+y)= sinx cosy + cosx siny
Sin difference of angles
Sin(x-y)= sinx cosy- cosx siny
Cos sum of angles
Cos(x+y)= cosx cosy - sinx siny
Cos difference of angles
Cos(x-y)= cosx cosy+ sinx siny
Tan sum of angles
Tan(x+y)= tanx + tany/ 1- tanx tany
Tan difference of angles
Tan(x-y)= tanx- tany/ 1+ tanx tany
Sin ratio of double angles
Sin2x= 2 sinx cosx
Cos ratio of double angles
Cos2x = cos squared x - sin squared x
= 1-2sin squared x
= 2 cos squared x - 1
Tan ratio of double angles
Tan 2x= 2tanx/ 1- tan squared x
Sin♎️ ratio in terms of tan♎️/2
Sin♎️= 2t/ 1+ t squared
Sin favourite so + bottom
Cos♎️ ratio in terms of tan♎️/2
Cos♎️= 1-t squared/ 1+ t squared
MISH MASH MIDDLE
but bottom still positive
Tan♎️ ratio in terms of tan♎️/2
Tan♎️= 2t/ 1-t squared
Tan is a weird minus bottom
Solving Further Trigonometric Equations
asin♎️ +/- bcos♎️ OR acos♎️ +/- bsin♎️
Sin further trig equations
1) asin♎️+bcos♎️= rsin(♎️+♉️)
2) asin♎️-bcos♎️= rsin(♎️-♉️)
Cos further trig equations
1) acos♎️+ bsin♎️= rcos(♎️-♉️)
2) acos♎️- bsin♎️= rcos(♎️+♉️)
Further trig equations r=
r= root a squared plus b squared
Further trig equations ♉️
Tan♉️= b/a
Solving further trig equations
1) convert the equation into asin or acos form
2) using this form find a and b and hence r and ♉️
3) convert to r sin/cos(♎️+♉️) form and use compound angles to solve
Sin general solution
If sin♎️=sin♉️, then
♎️=180n + ♉️(-1)^n
Cos general solution
If cos♎️=cos♉️, then
♎️=360n +/- ♉️
Tan general solution
If tan♎️=tan♉️, then
♎️=180n+♉️
Radians to degrees
multiply radians by 180/pi
Degrees to radians
Multiply degrees by pi/180
