How to Measure a Common Right Triangle
A right triangle is a triangle with an interior 90-degree angle and three sides, the longest of which is called the hypotenuse.The Pythagorean theorem defines the relationship of a right triangle's sides as a^2 + b^2 = c^2, where "a" and "b" are sides and "c" is the hypotenuse. The sides and non-90-degree interior angles can also be found using the trigonometric functions sine, cosine and tangent. The functions solve for an angle using the hypotenuse, the angle's closest side (adjacent) and opposite side (opposite).
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Instructions
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Find the length of one of the sides when given the two other sides by using the Pythagorean theorem. For example, if the hypotenuse is given as 4 and one side as 2, plug in the numbers to get 2^2 + b^2 = 4^2, which equals 4 + b^2 = 16. Subtract both sides by 4 to get b^2 = 12. Take the square root of both sides to eliminate the exponent: b = √12.
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Find the length of a specific side using trigonometric functions when not enough information is given to use the Pythagorean theorem. Determine which trigonometric function fits with the information you're given. For example, if a triangle has an angle of 23 degrees and a hypotenuse length of 5, use the sine formula, sin (23 degrees) = x/5, to find the opposite side's length. Solve sin (23 degrees) using a calculator and round to the nearest hundredth: 0.39 = x/5. Multiply both sides by 5 to find the side x: 1.95 = x.
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Find the value of one of the non-90-degree angles by using the inverse of a trigonometric function. Determine which function applies with your given information. For example, if a triangle has an unknown angle "x," an opposite side of 10 and an adjacent side of 8, use the tangent formula and plug in the given information: tan (x degrees) = 10/8, or tan (x degrees) = 1.25. Solve for x by using a calculator to find the inverse of tangent: x = tan^-1 (1.25) = 51.34 degrees.
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Tips & Warnings
The definitions of the trigonometric functions in terms of a right triangle are sin(x degrees) = opposite/hypotenuse; cos(x degrees) = adjacent/hypotenuse; and tan(x degrees) = opposite/adjacent.
When using a calculator to calculate an inverse function, hit the inverse key rather than the regular function key. Also check that your answers are set in degrees, not radians.
