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# Trig values of π/4

CCSS.Math:

## Video transcript

so we have depicted here is a unit circle centered at point a and the point B lies on the circle and then we drop the perpendicular front with from from point B to point D point D lies on the positive x-axis and they form this triangle abd and they tell us an angle b ad angle bad it has a measure of pi over 4 radians what i want to do in this video is use our knowledge of trigonometry and use our knowledge of triangles in order to figure out several things so the first thing we want to figure out is what's the measure of angle what's the Radian measure what's the Radian measure measure of angle abd of angle a B D actually let's just do that first and then I'll talk about the other things that we need to think about so I assume you've paused the video and tried to do this on your own so let's think about what abd would be we know two of the angles of this triangle so if you know two of the angles of this triangle you should be able to figure out the third now what might be a little bit unfamiliar we're used to saying that the sum of the interior angles of a triangle add up to 180 degrees but now we're thinking in terms of radians so we could say that the sum of the angles of a triangle add up to instead of saying 180 degrees 180 degrees is the same thing as PI radians so this angle Plus that angle Plus that angle are going to add up to PI so let's just say that this right over here let's say that let's just say measure of angle abd in radians plus PI over 4 plus PI over 4 plus this is a right angle what would that be in radians well the right angle in radians a 90-degree angle in radians is PI over 2 radians so plus PI over 2 when you take the sum of them the interior angles of this triangle they're going to add up to PI radians which is of course the same thing as 180 degrees and now we can solve for the measure of angle abd measure of angle a BD is equal to PI minus PI over 2 minus PI over 4 I just subtracted these two from both sides so this is going to be equal to let's see we could put a common denominator of 4 then this is 4 PI over 4 this is minus 2 PI over 4 and this of course is minus PI over 4 so 4 minus 2 minus 1 is going to get us to 1 so this is going to give us 2 PI over 4 so the measure of angle abd is actually the same as the measure of angle B ad it is PI it is PI over 4 so that angle right over there is PI over 4 now what does that help us with so if we know that this is PI over 4 and that is PI over 4 radians and once again we know this is a unit circle so we know the length of segment a-b that this which is a radius of the circle is the radius of the circle is length 1 what else do we know about this triangle can we figure out the lengths of segment ad and the length of segment dB well sure because we have two base angles that are that have the same measure that means that the corresponding sides are also going to have the same measure that means that this side is going to be congruent to that side I can reorient it in a way that it might be make it a little bit more a little easier to realize if we were to flip it over if we were to not completely flip it over but if we were to make it look like this so the triangle we could make it look like let me make it a little look a little bit a little bit more like this actually want to make it look like a right angle though so my triangle let me make it look like there you go so if this is D if this is D this is B this is a this is our right angle now this is PI over 4 radians and this is also PI over 4 radians when your two base angles are the same you know you're dealing with an isosceles triangle and isosceles and but because they're not all the same you know it's not equilateral if if all the angles were the same this would be equilateral but this is an isosceles Nonnie lateral triangle so if your base angles are the same then you also know that the corresponding sides are going to be the same these two sides are saying this is an this is an isosceles triangle and so how does that help us figure out the lengths of the sides well if you say that if you say that this side has length X and that means that this side has length X this side has length X then this side has length X and now we can use the Pythagorean theorem we could say that x squared this x squared plus this x squared is equal to the hypotenuse squared is equal to 1 squared or we could write the 2x squared is equal to 1 or that x squared is equal to 1 over 2 or just taking the principal root of both sides we get X is equal to 1 over the square root of 2 and a lot of folks don't like having a radical in the denominator they don't like having a rational number in the denominator so we could rationalize that I'm not the denominator by multiplying by the square root of 2 over the square root of 2 which would be C the numerator will have the square root of 2 and in the denominator we are just going to have square root of 2 times square root of 2 is just 2 so we've already been able to figure out several interesting things we're able to figure out a measure of angle abd in radians we were able to figure out the lengths of segment ad and the length of segment BD now what I want to do is figure out what are the sine cosine and tangent of PI over 4 radians given all of the work that we have done so let's first think about what is the sine let me do this in a color do it in orange given all the work we've done what is the sine of PI over 4 radians and I encourage you once again pause the video think about the unit circle definition of trig functions and think about what this is well the unit circle definition of trig functions in this angle this PI over 4 radians is forming an angle with the positive x-axis and where it's terminal ray intersects the unit circle the x and y coordinates of this point are what specify the cosine and sine so the coordinates of this point are going to be are going to be the cosine of PI over 4 radians is the x-coordinate the sine of PI over 4 radians is the y-coordinate sine of PI over 4 so what's the y-coordinate going to be if we want the sine of PI over 4 well it's going to be this length right over here which is the same thing as this length which is the length of X which is square root of 2 over 2 now what is the cosine of PI over 4 and once again I encourage you to pause the video to think about it well it's this x-coordinate what's the x-coordinate the x-coordinate is this distance right over here which is once again going to be X which is square root of 2 over 2 now what is the tangent of PI over 4 going to be well the tangent of PI over 4 is just sine of PI over 4 over cosine of PI over 4 now both of these are this exact same thing they're both square root of 2 over 2 so you can have square root of 2 over 2 divided by square root of 2 over 2 well that's just going to give you 1 and that also makes sense because remember the tangent of this angle is the slope of this line and we see the slope for every X we move in the horizontal direction we move X up so our change in Y over change in X is essentially x over X which is equal to 1