Math the Unit Circle

How can you figure out the secant of angles/radians?

you can first find it for it's normal form (sin instead of csc) and then flip it over. Or you can use it as a checking mechanism to make sure that the secant is correct. You can use find for example Secπ/4 by using a special triangle and finding the H/A instead of the A/H

What is 120 degrees in radians? 135?

2π/3 is 120 degrees 3π/4 is 135 degrees

What is the secant of Sinθ?

Cscθ It is the opposite of sin, so since Sinθ = O/H Cscθ = H/O

What are the special triangles?

It can be x but we just use 1 for the X

Where is the y-value for a point of inflexion?

It's on the original function not on the zero.

Does Pythagorean theorem work for special triangles?

Yes, there are still right triangles

0 divided by 1

0

sin(0)

0

cos(0)

1

derivative of √x?

1 / 2√x 1/2 (x)^-1/2 1/2 (1/√x)

things to remember for paper 1 (non-calc)

1. Show your working - detailed and clear 2. use formula booklet (start with formula) 3. if you finish early - go overrr 4. watch the simple math - don't make simple mistakes

What is lnx differentiated?

1/x not ln(1/x)

if 6 is equal to log3(p), what is log3(p)^2?

12 2 (log3(p)) 2(6)

if 6 is equal to log3(p) what is log3(9p)?

8 log3(9) + log3(p) 2 + 6

How would you solve derivative of Sin3x?

Chain rule! (Cos3x)(3)

What must you ALWAYS ALWAYS do?

Don't forget units!!!!!

What is the area of a sector formula?

It only works in radians!!! area of a circle is πr^2

What's important to remember for a show question?

Make sure the final answer is EXACTLY what the show asks for, ex. 8x +5 shouldn't be 5 +8x

When you have two variables to solve for, how do you do that?

Make two equations then set each equation equal to one of the variables then set the two equations equal to each other and solve.

What is the secant of Cosθ?

Secθ It is the opposite of Cos, so since Cosθ = A/H Secθ = H/A

How would you put sec into the calc to solve?

Since there are no secant buttons, for any secant, just do the reversed normal form. Ex. Secθ = 1/cosθ

How would you solve Sin 9π/4 ?

So we know that it's sinπ/4, which is positive 1/root 2. But know we need to know if it's negative because of being multiplies by 9.

What does Tanθ =

So you can solve this way to find a tan point on circle

sectors

The arc is the curved part The radius is the lines to the centre

what are does d^2y / dx^2 mean?

The second derivative

How do you find the HA?

Use the limit! lim X--> infinity

What is the trig identity that makes the Pythagorean theorem with the Sin and Cos identities?

You can use this indentity to replace any cos^2 or sin^2 you have in an equation with the other part of the identity. ex. cos^2 = 1- sin^2 so therefore (1 - sin^2) can be put in an equation

If you get an answer like 5=5 that means

all real solutions.

What should you do for the calculator part?

always try to solve with he calc not algebraically if you can (graph)

when you have a triangle and you are looking for a side but you don't know for sure which is the hypt, opp, and adj, what do u do?

dont use sin, tan or cos to solve, just the sin or cos formulas

What is the derivative of e^x

e^x

Inverse of f(x) = (x-5)^3?

f'(x) = 3√x +5

how could you find TWO solutions for sinx = 2/3?

graph and and find the two intersections, using inverse of sin in calc will only give u 1

where is the angle of depression?

it's not in the triangle

wx^2 means that...

just the x is squared!

If you get cosx = √-5

no solutions

can you find the derivative of something like 1/x^3 for just the denominator while leaving it under 1?

no you must always first make it x^-3 then find the derivative

Do you simplify an algebraic expression if youre gonna graph it?

no!!! (don't cancel negatives or do anything like that if you're trying to find the intersect)

What is the normal again?

the negative reciprocal y = - 1/f'(a) x + b

When you have degrees in an equation when trying to find something about a sector what do you do?

turn it to radians! (degreesπ / 180)

1 divided by 0

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if we know that f(x) = x+5 and we know that f(g(x)) = 8x2, how do we find g(x)?

we need to set it up like g(x) +5 = 8x2 then solve for g(x)

what is the the VA of 2ln(x-3)

x =3

what is 3√x^6 (the 3 is not being multiplied it's cube rooted)

x^2

What is the circumference of a sector formula?

Arc length = θr (again only in radians!!) Circumference of a circle is 2πr

What is the way to show HA? VA?

HA: y= 2 VA: x=2

how would you solve a problem that says "find the number of shirts when the cost is lowest?" and you have equation of cost for shirts

You find the derivative then set it equal to zero since that's the minimum you'll get

What must you do once you get answers?

You have to plug in your values back into the equation to see which is correct because by squaring/dividing by cosx or sinx you get an invalid solution.

What is a unit circle?

a circle with a radius of 1

When is 2π or any other value added to a radian to get it into the right domain?

it is added at the end!

What happens if you have an equation cos(4x) - 3cos^2(2x) = 4

let A = 2x cos(2A) - 3cos^2(A) = 4 (then expand cos2A and cancel like terms

If an cosx is an obtuse angle, is cosx going to be positive or negative

negative because obtuse means greater than 90 degrees so in the 2nd or 3rd quadrent, and that's where cos is negative (not the fourth quadrent because that makes an acute angle again)

What is the secant of Tanθ?

Cotθ It is the opposite of Tan, so since Tanθ = O/A Cotθ = A/O

How would you solve Sinx - Cos(2x) = 0

It's important to do the whole "Let A = __ because that helps make the equation much simpler (it can also be Let A = 2x, so that you deal with the 2 later

What is the long way for converting degrees to radians

Proportion π/180 = x/degrees

things to remember for paper 2 (calc)

1. very little writing - not a lot of algebra--> USE CALC 2. Be sure in right mode - u switch back between the two 3. remember units 4. x-coordinate is just x value (coordinate is both), or max value is just y --> reread the question so u answer it correctly 5. cross out wrong answer after factoring

Given that cosA=5/6 find the value of SinA

Use the pythag identity Sin^2 x + Cos^2 x = 1 1- (5/6)^2 = Sin^2 x 11/36 = sin^2 x Then square root both sides and there you go

How do you find the derivative of f(g(x)?

Using chain rule!

The four quadrants of the unit circles and their + / -

All Students Take Calculus In A, the first quadrant, Sin, tan and Cos are all positive In S, the second quadrant, only Sin is positive In T, the third, only Tan is positive In C, the fourth Quadrant, only Cos is positive

sinθ graph

All the points are from the unit circle, and you can get the points in between from the special triangles ex. sin3π/4 --> π/4 --> 1/2 and 3π/4 is in the second quadrent so that's positive for sine

What do you need to do if you are made the domain smaller like with sin2x = 1/2

Always include all the values for that radian (pi/6) in the domain even if it's negative then you cross out ones, because there are some values that when you divided pi/6 by 2 to get to x they are small enough to be in the positive quadrent. Sin(2x) makes the graph fit twice as much into the same domain so more radians fit into each quadrent so there may be more positive ones (so you would include pi/6, 5pi/6, 7pi/6, 11pi/6 then didive by 2 THEN cancel out the ones in negative places)

How would you solve: Given that Sinx = 1/3, find the value of Cos4x

Cos4x = Cos2(2x) Cos2x = 1 - 2sin^2x Cos2x = 1 - 2(1/3)^2 Cos 2x = 7/9 Then put back into first equation let W = 2x Cos4x = Cos2(W) where cosw = 7/9 Cos4x = 2Cos^2w -1 Cos4x = 2(7/9)^2 - 1 Cos4x = 17/81

THE DIFFERENCE BETWEEN INVERSE AND DERIVATIVE

Inverse is f^-1(x) while derivative is f'(x) DON'T MIX THEM UP

What happens in the degree/radian is negative? ex. Sin(-π/4)

It goes the opposite direction around the circlE

What is the discriminant used for?

It is used to find the number of solutions For only 1 solution : = 0 For 2 solutions: >0 For no solutions: <0

What is the derivative of 1/4x^4 ?

It's -x^-5 you have to separate the 1/4 from the 1/x^4 since THE EXPONENT DOES NOT APPLY TO THE 4, JUST THE X, so then so find the derivative of the x then multiply the 4 back in and they cancel. THIS IS REALLY IMPORTANT, remember that unless there are parentheses, the exponent only applies to the x so you take out the coefficient

Where is the reference line on the unit circle?

It's the x-axis

For a situation like 2SinxCosx = Sinx, is it fine to divide sinx from both sides?

No because this cancels out a possible solution which would have been sinx=0, so generally don't cancel out by sinx,cosx,tanx, x, logx, etc. (Instead take the Sinx to the other side and take out the Sinx from both parts as a common factor.)

Can you use special triangles to find an angle for 90 degrees?

No, only for the other angles. For angles on the axises like 90, 180, 270 or 360 degrees, just use the visualization of the unit circle or the sin/cos graph to find out what its value is. Ex. Sin90 = 1

How do you find the gradient using the derivatives?

Setting the derivatives equal to each other find intersect put the x point of that intersect back into one of the derivatives to find the gradient

If you have something like 11π/3 which ends up on the 360 point, which quadrent does that technically belong to for figuring out the sign of it?

To figure out if a radian/angle is positive or negative and it lands on one of the axises, use the quadrant it just passed. So you would put 360 degrees into the fourth quadrent where only Cos is positive.

How do you show if something is a minimum or maximum?

Use f'(x) > 0 --> f(x) is increasing f'(x) < 0 --> f(x) is decreasing Use this to see the slopes of the original function around the max/min then use that to show its a max/min

Domain change?

When you have an equation like Sin2x = 1 and the domain is between 0 and 2π, the domain HALVES since it is 2x, and if it's x/2 then it DOUBLES, so the amount of values for 2x would be between 0 and π But the better way to solve it is just use the expanded double angle formula for 2x

what is (x,y) in terms of the unit circle?

X tells you the distance from the y-axis and y tells you the height from the x axis, how high the spoke is (x,y) --> (cosθ, sinθ) (cosθ, sinθ) shows any point on the unit circle The spoke always has a length of 1 (the picture)

How would you solve something that says that " if f(x) = m, what are the values for m in which f(x) has 4 solutions? " and we had a graph of f(x)

m is whatever the equation f(x) is equal to, so it's like the y value. Therefore, the y value that has 4 corresponding x values is a range on the graph. On this graph, its 1 < m < -2

Double Angle Formulas

sin(2x) = 2sin(x)cos(x) don't multiply the x by 2, expand it like this (also the picture) All in the in formula booklet, but you can choose any of the Cos formulas depending on the one that is most appropriate for the equation (to have on sin or only cos)

How would you solve? The line y = kx − 5 is a tangent to the curve of f. Find the values of k. we know that f'(x) = -2x+5

so we know that k = -2x+5 since the slope = derivative so we put -2x+5 into y = kx − 5 and solve for x and then plug x back in

How do you solve something that is like Cos^2(60)

you solve cos(60) then square the answer

theta symbol

θ

What is 30 degrees in radians? What is 60 degrees? What is 45 degrees?

π/6 is 30 degrees π/3 is 60 degrees π/4 is 45 degrees