The arc length is based off the circumference of the circle of which is two pi r. If we think about wanting an arc length, we only want a portion of the full circle. so we want a proportion of it. Let's say we want theta as the angle so if we want the arc length, the letter that's usually used is S, equal we could think of that theta out of a whole circle, and if we are doing this in radians, the full circle in radians would be two pi, so we want a ratio, we want a ratio, we want our theta out of of two pi and then, that is going to be multiplied by the circumference of the full circle, because we're wanting a ratio or a portion of the full what happens next is the two pis cancel. So S equals theta r, if we're in radians that's the easiest equation Now, if any unit circle we know that in a unit circle our r equal one so thus S equal theta in a unit circle so the angle measure equals the arc length in a unit circle terminal rays and initial rays. The initial ray is usually thought of as the x-axis the terminal ray is where the ray actually ends, or the angle ends. Positive angles go counterclockwise, negative angles go clockwise our six trigonometric functions. We have sine being opposite over hypotenuse, cosine being adjacent over hypotenuse tangent being opposite over adjacent Cosecant: hypotenuse over opposite. Secant: hypotenuse over adjacent Cotangent: adjacent over opposite so sine theta is just one over secant and cosecant is one over sine. They're reciprocal functions of each other cosine is one over secant and secant is one over cosine those, once again, reciprocal functions tangent theta equal one over cotangent, which is also sine over cosine cotangent is one over tangent or cosine over sine. Once again, reciprocal functions we're going to look at some standard values and we get our standard values based off of a 45/45 right triangle and also a 30/60 right triangle if we start with the 45/45 right triangle we know that the two sides opposite the equal angles have to be equal. So we can call those two sides one now by pythagorean theorem, we can find the hypotenuse because a squared plus b squared equal c squared in a right triangle and we know that a is one squared and b is one and we're going to square it and to find the c, we're going to square root so this hypotenuse is going to be square root of two now if we do our trig functions, we can do sine of 45 which would be one over root two. To rationalize, we're going to multiply the top and the bottom each by root two so we get square root of two over square root of four or square root of two over two cosine 45 is gonna be the same thing. One over root two, which simplifies to root two over two tangent 45 opposite over adjacent, or one secant 45 It's going to be root two over one, or just root two cosecant 45: root two over one again and cotangent 45 one over one now we also know that 45 degrees is really the same thing as pi fourth radians in our 30/60 right triangle, if we thought about starting with an equilateral triangle and letting each side be a length of two then when we dropped this altitude or or the height we split the 60 degree into 30 and 30, so we had to split the base into one and one if we do that we can now find the height based off the pythagorean theorem. So the square root. The hypotenuse here was the two, so tow squared minus one squared or square root of three so if we want to do sine of 60 degrees that's the opposite over hypotenuse cosine of 60 would be adjacent over hypotenuse tangent 60 opposite over adjacent, so root three over one which is just root three cosecant of 60 is going to be two root three over three if we rationalize secant of 60 is two and cotangent of 60 is root three over three if we do the 30 degree, sine of the 30 degree: sine is one half cosine of 30 degrees is root three over two tangent of 30 degrees is root three over three cosecant of 30 degrees is two secant of 30 degrees two root three over three and cotangent of 30 degrees root three we know that 60 degrees is pi thirds and we know that 30 degrees is pi sixths if we forgot how to convert degrees to radians think about taking whatever your degrees is, 60 degrees, and multiply it by the fact that 180 degrees is the same thing as pi radians so 60 and 180 reduce to one third times the pi we have also to consider the quadrants that things are positive and negative in. In a quadrant, in a cartesian plane, all of the trig functions are positive in the first quadrant so positive, positive, positive the cosine is usually thought of as our left and right the sine are up and down and then the tangent is just the ratio of the sine divided cosine. So if we look here in the second quadrant, we've gone left, so that cosine is negative, we've gone up so the sine is positive and a positive divided by negative is negative in the third quadrant we've gone left, so negative. We've gone down, negative, and a negative divided by a negative is a positive in this last quadrant: cosine, we've gone right, so positive. And we've gone down so negative, and negative divided by positive is a negative so one way to remember this is all students take calculus all of them are positive in the first, sine is positive in the second tangent positive in the third cosine positive in the fourth, and that also includes the reciprocal functions periodic functions are functions that repeat. So if we have sine of theta plus two k pi that's going to really equal sine theta, because we know that the sine values repeat every two pi where k is just some integer cosine repeats every two pi also, so cosine of theta plus two k pi is really cosine theta tangent repeats every pi. So tangent of theta plus k pi is really tangent theta when we look at a unit circle we know that the r equals one, but if we looked at any circle and put in a right triangle we could identify the theta, the x, the y, and the r on our right triangle. x being the distance left and right, y being up and down, r being the radius so cosine theta is adjacent over hypotenuse or x over r, so x equals r cosine theta sine theta equal y over r, or r sine theta is y. Now in a right triangle, we know that the two sides squared, the sum of them, equals the hypotenuse squared so x squared plus y squared equal r squared. But in this case, we showed that x was really r cosine theta, and we're going to square that y was r sine theta we're going to square that and that's going to equal r squared. So we get cosine squared theta plus sine squared theta equaling one if we divided everything through by an r squared that's a pythagorean identity. We have two more pythagorean identities that come relatively easy from starting with this one and dividing everything through by cosine squared for each of those three terms and then we're going to do the same thing by dividing everything through by sine squared. Cosine squared over cosine squared is one sine over cosine is tangent, so one plus tangent squared theta equal one over cosine squared theta is secant squared theta so if we did the same thing but divided every term by sine squared theta we'd get our third pythagorean identity and our third pythagorean identity says cotangent squared theta plus one equal cosecant squared theta. So those are our three pythagorean identities. Cosine squared theta plus sine squared theta equal one one plus tangent squared theta equal secant squared theta data and cotangent squared theta plus one equal cosecant squared theta addition and subtraction formulas: cosign alpha plus or minus beta is cosine alpha cosign beta minus plus sine alpha sine theta when we've read this we need to think about the top sign goes with the top sign as we read across, the bottom sign is with the bottom sign sine alpha plus or minus beta is sine alpha cosine beta plus minus cosine alpha sine beta tangent alpha plus or minus beta is tanget alpha plus or minus tangent beta over one minus plus tangent alpha tangent beta the double angle formulas. Cosine two theta: cosine squared theta minus sine squared theta, which is also equal to two cosine squared theta minus one we can get that one by using the pythagorean identity we just developed and replacing sine squared theta as one minus cosine squared theta those are both also equivalent to one minus two sine squared theta sine two theta is two sine theta cosine theta tangent two theta is two tangent theta divided by one minus tangent squared theta half angle formulas. Cosine squared theta is one plus cosine two theta over two sine squared theta is one minus cosine two theta over two. Tangent squared theta, one minus cosine two theta over one plus cosine two theta the law of cosines. c squared equal a squared plus b squared minus two a b cosine C law of sines, sine A over a equals sine angle B over b equals sine angle C over side c transformation of a graph. If we have y equal a times the function of the quantity b times the quantity x plus c plus d the a in front tells us a vertical stretch or compression the b tells us a horizontal stretch or compression the c tells us the horizontal shift and the d tells us the vertical shaft the c is actually needing to be taken into account with the b, because the horizontal stretch and compression is going to affect the horizontal shift also okay, we're going to develop a couple special inequalities that we'll use. SO we're going to start with the fact that we know that S equals r theta and in a unit circle r equal one. So we know that the arc length is really the same thing as the angle measure that's going to be important because now we're going to look at the fact that PQ is really just sine theta and OQ is really just cosine theta so if I wanted QA QA would be the whole distance of one minus the cosine theta now we see that it's a right triangle, so we know that PQ squared plus QA squared has got to equal line segment PA squared well PQ squared is really just sine squared theta and QA squared is just gonna be one minus cosine theta squared and and PA squared we're going to leave as PA the line segment PA squared now we know that the line segment PA has gotta be less than the arc length PA. And the arc length PA is equal to theta so we know that the line segment PA has to be less than theta so we know that sine squared theta or a piece of this addition problem has to be less than theta squared we also know that one minus cosine theta squared would have to be less than theta square, because the two had to add up together to give us something that was smaller still than the theta to get rid of the sine squared, we're going to take the square root so we're going to have negative the absolute value of theta less than sine theta less than the absolute value of theta doing the same process for the other equation we're going to get negative absolute value of theta less than one minus cosine theta less than the absolute value of theta