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Lesson 157: Special Reference Right Triangles
In this lesson we'll learn about some special right triangles that come up frequently.
A very common right triangle has degree measurements of 45-45-90. We know that both legs will be equal, since they are opposite angles of equal measure. Let's say that each side measures 1 unit. It's easy to use the Pythagorean Theorem to figure out that the hypotenuse of such a triangle measures √2.
It's
important to memorize this relationship in a
45-45-90 triangle. Understand that if we multiply each side by
a constant, and we'll still have a 45-45-90 triangle. For
example, a right triangle with measurements of 7, 7, 7√2 is still a
45-45-90 triangle. All we did was multiply each side of the
reference triangle by 7.
By memorizing this reference triangle, it is very easy to compute the sine, cosine, and tangent of 45 degrees. For example, we can see that sin (45), as well as cos (45) both equal 1/√2. We can see that tan (45) = 1.
Another reference triangle can be formed by first examining an equilateral triangle with sides measuring 2. Of course each angle measures 60 degrees. In the diagram, an altitude has been dropped from the upper vertex which bisects the base, as well as the upper vertex angle. You can now see that two right triangles have been formed, each with a base of 1, and a hypotenuse of 2.
Using
the Pythagorean Theorem, we can easily compute that the other leg
(and the altitude of the original triangle is √3. You can also
see that what we are working with is a 30-60-90 triangle.
Since √3 is bigger than 1, we know that it must lie opposite the 60
degree angle, which is larger than the 30 degree angle.
This is another important relationship to memorize. Remember, if we multiply each of the sides by a constant, it will not change the angle measurements of the triangle, and the relationship will still hold. Knowing this, we can easily compute the sine, cosine and tangent of 30 and 60 degree angles whenever we need to.
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