Trigonometry and Wave Function Flashcards
π radians =
180 degrees
30 degrees in radians
π/6 radians
45 degrees in radians
π/4 radians
60 degrees in radians
π/3 radians
90 degrees in radians
π/2 radians
135 degrees in radians
3π/4 radians
360 degrees in radians
2π radians
how to convert from degrees to radians
multiply by π and divide by 180
how to convert from radians to degrees
multiply by 180 and divide by π
describe exact values triangle for 45 degrees
right angled triangle
45 degrees at each of other vertices
root 2 on hypotenuse
1 on other sides
describe exact values triangle for 30/60 degrees
right angled triangle
1 along base
2 on hypotenuse
root 3 on other side
60 in angle between 1 and 2
30 in angle between 2 and root 3
how to work out exact values for 0,90,180,360 etc.
trig graphs
when would there be no solutions for sin/cos
has to be greater than or equal to -1 and less than or equal to 1
what does sin2x mean?
there are 2 cycles of sin graph in 360 degrees
what part of the question should you pay close attention to when deciding how many answers are required?
the domain
if there is a squared term what should you remember
that it could be positive or negative when you square root so will need to work out solutions in all quadrants
what should be noted if an angle is acute?
a cast diagram is not necessary
what is the result when we add or subtract angles?
a compound angle
sin (A+B) =
sinAcosB + cosAsinB
sin (A-B) =
sinAcosB - cosAsinB
cos (A+B) =
cosAcosB - sinAsinB
cos (A-B) =
cosAcosB + sinAsinB
how to layout question asking you to prove one of the compound angle formulas using given numbers
work out LHS and RHS separately then state “since LHS=RHS, the claim it true for…”
how to find the exact value of sin75
sin(45+30)
if a question gives you values for sin p and sin q, how do you calculate sin(p+q)?
- draw triangles out (sin=o/h)
- use pythagoras to work out other side and therefore the values of cos p and cos q
- use compound angle formula
sinx / cosx =
tan x
sin squared x + cos squared x =
1
1 - cos squared x =
sin squared x
1 - sin squared x =
cos squared x
double angle formula for sin2A
2sinAcosA
double angle formula for cos2A
cos squared A - sin squared A
2cos squared A - 1
1 - 2sin squared A
further trig equations: how to use double angle formula to solve trig equations
- replace 2A term using double angle formula
- make sure every term is on one side and equation is equal to zero
- factorise the expression and then solve each part
wave function: how to work out k
square root of p squared + q squared
wave function: how to work out tan a
ksina / kcosa
simple wave function question steps
- expand kcos(x-a) using compound angle formula
- rearrange so you can compare with pcosx + qsinx
- state values of ksina and kcosa
- CAST diagram
- find k and a
- state pcosx + qsinx in the form kcos(x-a) using these values
steps for wave function with multiple angles
exactly the same as before but use kcos(2x-a) or whatever number/form in question
what are the maximum and minimum values of the sine and cosine functions?
maximum 1
minimum -1
how to work out maximum value of a trigonometric function
- follow wave functions steps
- max value of k occurs when kcos(x-a)=k
- rearrange to find x
how to solve equation that involves both a sin(nx) and a cos(nx) term
- wave function steps as normal
- equal kcos(x-a) to RHS
- rearrange and use CAST diagram to solve for x (answers in suitable quadrants and double check the domain)
how to sketch the graph of y=pcosx+qsinx
- write in form kcos(x-a)
2. k is amplitude and a is the phase shift (check if there is also a shift up/down)
