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CCSS.Math: ,

let's say that we're told that some angle theta which is going to be expressed in radians is between negative 3 PI over 2 and negative PI it's greater than negative 3 PI over 2 it's less than negative PI and we're also told that sine of theta is equal to 1/2 from just from this information can we figure out what the tangent of theta is going to be equal to and I encourage you to pause the video and try this on your own in case you're stumped I will give you a hint you should use the Pythagorean identity the fact that sine squared theta plus cosine squared theta is equal to 1 so let's do it so we know the Pythagorean identity sine squared theta plus cosine squared theta plus cosine squared theta is equal to 1 we know what sine squared theta is sine theta is 1/2 so this could be re-written as 1/2 squared plus cosine squared theta plus cosine squared theta is equal to 1 or we could write this as 1/4 plus cosine plus cosine squared theta is equal to 1 or we could subtract 1/4 from both sides and we get cosine squared theta is equal to C you subtract 1/4 from the left hand side then this 1/4 goes away that was the whole point 1 minus 1/4 is 3/4 is 3/4 so what could cosine of theta be well when I square it I get positive 3/4 so it could be the positive or negative square root of 3/4 so cosine of theta cosine of theta could be equal to the positive or negative square root of 3 over 4 of 3 over 4 which is the same thing as the positive or negative square root of 3 of 3 over the square root of 4 which is 2 so it's a positive or negative square root of 3 over 2 but how do we know which one of these it actually is well that's where this information becomes useful that's where this information actually becomes useful let's draw our unit circle if you're saying well why am I even worried about cosine theta well if you know sine of theta you know cosine of theta tangent of theta is just sine of theta over cosine theta so then you will know the tangent of theta but let's figure out let's look at the unit circle to figure out which value of cosine we should use so let me draw it the unit circle that's my y-axis that is my x-axis and I will draw the unit circle in pink so that's my best attempt at drawing a circle please forgive me for its lack of perfect roundness and it says theta is between it's greater than negative 3 PI over 2 so where's negative 3 PI over 2 so let's see this is negative PI over 2 so this is one side of the angle let me do this in a color so let's see this one side of the angle is going to be along the positive x axis and we want to figure out what the other side is so it's going to this this right over here that's negative PI over 2 this is negative pi so it's between negative pi which is right over here so let me make that clear negative pi is right over here it's between negative pi and negative 3 PI over 2 negative 3 PI over 2 is Right negative 3 PI over 2 is right over here so the angle our angle theta our angle theta is going to put us someplace someplace over here someplace over here and the whole reason I did this so this whole arc right here you could think of this as as the measure of angle theta right over there and the whole reason I did that is to think about whether the cosine of theta is going to be positive or negative we clearly see it's in the second quadrant the cosine of theta is the x coordinate of this point where our angle intersects the unit circle so this point right over here this right over here actually let me do it in that orange color again this right over here that is the cosine of theta now is that a positive or negative value what's clearly a negative value so for the sake of this example R cosine theta our cosine theta is not the positive one it is the negative one so we can write that cosine theta is equal to the negative square root of three over two so we figured out cosine theta but we still have to figure out tangent of theta and we just have to remind ourselves that the tangent of theta is going to be equal to the sine of theta sine of theta sine of theta over the cosine of theta over the cosine of theta well they told us the sine of theta is one half so it's going to be one half over cosine of theta which is negative square root of three over two negative square root of three over two and what does that equal well that's the same thing as one half times the reciprocal of this so times negative negative 2 over the square root of three these twos will cancel out and we are left with negative 1 over the square root of 3 now some people don't like a radical in the denominator like this they don't like an irrational denominator so we could rationalize the denominator here by multiplying by square root of three over square root of 3 square root of 3 over square root of 3 and so that gets us this will be equal to negative square root of 3 over 3 is the tangent is the tangent of this is the tangent of this angle of this angle right over here and that actually makes sense because the tangent of the angle is the slope of this line and we see that it is indeed a negative a negative slope