Special Right Triangles

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Trigonometry: Special Right Triangles

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Key Questions

  • There are 2 types of special right triangles.
    Type 1. Triangle that is half of a equilateral triangle. Its 3 angle measures are: 30, 60 and 90 deg. Its side measures are : a, a/2; and (a*sqr.3)/2.
    Type 2. Triangle that has its side measures in the ratio of 3:4:5. The proof is given by the Pythagor theorem: c^2 = b^2 + a^2.
    Use of special right triangles.
    In the old time, people use the special right triangles with sides ratio 3:4:5 to figure out, in the field, a right angle or a rectangular, or square, shape.
    Now, students just use the properties of special right triangle to find, by computing, the unknown sides or angles.

  • The basic properties are: An interior angle of #45^@#, an exterior angle of #135^@#, an opposite angleof #45^@#, a right angle (#90^@#), opposite side length: #sqrt2/2#, and adjacent side length of #sqrt2/2#, and a hypotenuse of 1. (side lengths for a standard triangle inside of the Unit Circle)

    Explanation

    All standard right triangles inside of the unit circle have a hypotenouse of 1. This is because the unit circle is defined as a circle with a radius of 1. Thus we create Unit Circle Triangles by picking a radius, and drawing its vector components.

    In this case, we know we have a triangle with a primary angle of #45^@# and, as mentioned before, a hypotneouse of 1. As such we can use #sintheta# and #costheta# to determine the other two lengths.

    #sintheta=o/h->hsintheta=o->1*sin(45^@)=sqrt2/2#

    #costheta=a/h->hcostheta=a->1*cos(45^@)=sqrt2/2#

    Image from MontereyInstitute

  • #mathbf{30^circ"-"60^circ"-"90^circ}# Triangle

    The ratios of three sides of a #30^circ"-"60^circ"-"90^circ# triangle are:

    #1:sqrt{3}:2#


    I hope that this was helpful.

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