| Special Right triangle 45 degree #2 |
Explanation: Each of these images explain the relation between the unit circle in which we have been reviewing and the special right triangles that come into play. In example 1, it is shown that when you take the special right triangles such as the 45-45-90 and the 30-60-90 ones when you solve for the hypotenuse as being 1 then you are able to get your x and y values. These x and y values differ as you move into the different angles such as a 45 degree angle. When knowing that in this type of special triangle you know that the value of x and y will be the same which leaves the hypotenuse with whatever is the value of the sides and radical 2. With this knowledge in the unit circle when solving for the x and y value you have to divide each side by radical 2 and then simplify which leaves you with x= rad2/2 and y=rad2/2. This solution then allows us to connect the unit circle with this special right triangle because it ends up being the point, or coordinates for that 45 degree angle in the first quadrant as for the rest except in the other quadrants the sign change would occur. As for the next type of special right triangles from example 1 and prior knowledge we know that the leg across from 60 degrees is going to be the longest leg which would makes the leg rad 3 x, the shortest leg is the one that is across from 30 degrees and that will just remain as 1 or x, it is 1 in this example.the hypotenuse is shown as having a value in 2 in this triangle which needs to change to equalling 1, so to do so you divide each of the sides by 2. When doing so, you then get the value of x and y where x will stand for the shortest side being 1/2 and y (longest side) with the value of rad3/2. After finding these two values, we are able to connect these two values as equalling the coordinates which lie on the unit circle on the 60 degree angle, forming a special right triangle (30-60-90). This same process occurs when the triangle lies down and now the triangle is drawn on the unit circle with the triangle pointing up towards the 30 degree angle which would only switch the x and y value from the previous triangle explained. It would leave you with x(now being the longest side) equalling rad3/2 and y(the short side) equalling 1/2. This relationship between the unit circle and these special triangles is shown in the last 3 examples where the triangles are shown to be drawn on the unit circle. Overall, if they are drawn on the unit circle and are solved for with the hypotenuse equalling 1 then the values found for the sides will equal the coordinates for that degree (special right triangle degree).
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