How to Find an Angle in a Right Triangle If You Don't Have Any Angles
Trigonometry can take the unknown out of geometry. One place where this is particularly evident is in the analysis of right triangles. Right-angled triangles are characterized by a 90-degree angle. No more than two sides of the right triangle may be equal in length, and the side lengths can be predicted using the Pythagorean theorem. If any two of the side lengths of a right triangle are known, trigonometry can take advantage of the presence of the right angle to determine one or both of the other angles in the triangle.
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Instructions
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1
Label one of the two unknown angles as "X." Note that one of the three angles in a right triangle is always known, and the size of that angle is 90 degrees. The 90-degree angle should never be labeled as X.
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2
Label the longest side of the triangle as "H." This is the side positioned opposite to the right- or 90-degree angle of the triangle.
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3
Label the side opposite the angle X as "O."
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4
Label the side adjacent to angle X as "A." Note that side H (the hypotenuse) is also adjacent to angle X, but has already been labeled.
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5
Calculate the angle X using the trigonometric relationships for right triangles. X is given by the formula: X = arctan (O/A) = arcsin (O/H) = arccos (A/H). For example, a right-angled triangle with H = 5 cm, O = 4 cm, and A = 3 cm has an angle X (the angle between sides H and A) = arctan (O/A), which equals arctan (4 cm/3 cm) = 53.1 degrees. Alternatively, X = arcsin (O/H), which equals arcsin (4 cm/5 cm) = 53.1 degrees. X can also be calculated using X = arccos (A/H), which equals arccos (3 cm/5 cm) = 53.1 degrees.
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6
Repeat steps 1 to 5 to find the other unknown angle in the triangle.
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References
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