How to Find the Angles of a Right Triangle

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All triangles are marked by the same features: three sides and three angles. Right triangles are identified as such because one angle is measured at a perfect 90 degrees. Several methods may be used to find the other angles.

When Two Angles Are Known

  • All three angles of a triangle have a combined sum of 180 degrees. Aside from the right angle, which is 90 degrees, knowing another angle makes discovering the unknown as simple as subtraction. The unknown angle is the difference of 180 degrees, 90 degrees and the other known angle.

    For example:
    If presented with a right triangle and one of the angles is 37 degrees, the unknown will be 180 minus 90 (the right angle), minus 37 (the known angle), leaving a difference of 53 degrees for the unknown angle.

When Two Sides Are Known

  • If the lengths of two sides of a triangle are presented, either of the two unknown angles can be calculated using the trigonometric identities, Sine, Cosine and Tangent. These are special numbers that represent a conversion of ratios to angles in degrees based on a 360-degree circle where the hypotenuse is the radius, and the height and base represent the slope of an angle (height over distance or "rise over run").

    The acronym SOHCAHTOA helps you remember when to use which identity based on which sides are known.

    Sine (S) is used when the lengths of the known sides are Opposite (O) of the angle and the Hypotenuse (H), which is the longest side.

    If the length of the side opposite the unknown angle is 6 and the length of the hypotenuse is 8, the Sine ratio of the angle is 6 / 8. Or Sin(X) = (6/8).

    Cosine (C) is used when the lengths of the Adjacent (A) side and the Hypotenuse (H) are known.

    If the length of the side adjacent to the unknown angle is 3 and the length of the hypotenuse is 5, the Cosine ratio of the angle is 3 / 5. Or Cos(X) = (3/5).

    Tangent (T) is used when the Opposite (O) side and the Adjacent (A) side are known.

    If the length of the side opposite the unknown angle is 2 and the length of the side adjacent to the unknown angle is 3, the Tangent ratio of the angle is 2 / 3. Or Tan(X) = (2/3).

Solving for Sine, Cosine and Tangent

  • A graphing calculator is required to make use of three trigonometric identities. Once the ratio is determined, the equation can be put into a graphing calculator using the inverse identities: arcsin, arccos and arctan. These functions are represented as SIN^-1, COS^-1, and TAN^-1. Before continuing, make sure the calculator is set to DEGREE mode and not RADIAN.

    The previous examples can be solved by typing them into the calculator in the following format:

    SIN^-1(6/8)

    This results in about 49 degrees.

    COS^-1(3/5)

    This results in about 53 degrees.

    TAN^-1(2/3)

    This results in about 34 degrees.

Finding an Angle Using a Protractor

  • A protractor is a clear, plastic semicircle with notches numbered from 0 through 180. The numbers run both ways, but since the unknown is part of a right triangle, the angle must be smaller than 90 degrees. Along the bottom is a black line used to run along the base of the triangle. The point of the missing angle should be placed at the black line in the center. To find the missing angle, trace the hypotenuse to the numbered notch at the end and use the smaller of the two numbers.

References

  • Photo Credit yuriz/iStock/Getty Images
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