Unit 6, Lesson 2
Revisiting Right Triangles
Integrated Math 3
2.1
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Find , , and for triangle .
If the coordinates of point are , what is the value of , , and for triangle ? Explain or show your reasoning.
What are , , and ? Explain how you know.
Here is a triangle similar to triangle .
Here is another triangle similar to triangle .
In this triangle, the cosine of angle is the ratio of the length of the side adjacent to angle to the length of the hypotenuse, or . The sine of angle is the ratio of the length of the side opposite angle to the length of the hypotenuse, or . The tangent of angle is the ratio of the length of the side opposite angle to the length of the side adjacent to angle , or .
Now consider triangle , which is similar to triangle with a hypotenuse of length 1 unit. Here is a picture of triangle on a coordinate grid:
Since the two triangles are similar, angles and are congruent. So how do the values of cosine, sine, and tangent of these angles compare to the angles in triangle ? It turns out that since all three values are ratios of side lengths, , , and .
Notice that the coordinates of are because segment has length and segment has length . In other words, the coordinates of are .