SSA can sometimes be ambiguous because on some occasions, you can get multiple triangles when solving. You can't use the law of Sines at all times because you would only be finding one solution when a triangle can have two solutions or none. Although it would help you find out that there was no solution. This picture provides a good example for this theory. The side lengths that are 6 inches long are given. Now, using the law of sines you solve for the imaginary line "h" That would be directly in between those two lines, and would make the triangle a right triangle. You must first solve for h because that tells you whether or not you have solutions. If the given value for c is greater than h but less than b, you have two solutions because you can create two triangles. If the given value is the same as h, you have one solution and a right triangle. If the given value is less than h, you have one obtuse answer because you need a longer side length to create a second, acute triangle. If c is greater than b, then you can only have one solution, because the second one wouldn't fit in the other side of the triangle.
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Step 1: Expand the tan^2x to sin^2x / cos^2x Step 2: Expand the 1 to cos^2x / cos^2x. This can be done because cos^2x / cos^2x = 1 for all real numbers, or fake numbers, who knows? Step 3: Group the bottom sin^2x / cos^2x and cos^2x / cos^2x to make it into (sin^2x + cos^2x) / cos^2x. Step 4: Change (sin^2x + cos^2x) to 1 because (sin^2x + cos^2x) = 1 with the Pythagorean identity. Step 5: Cancel out the top cos^2x with the bottom cos^2x to leave just sin^2x. From this project, I learned valuable skills on working through a trig identity. I learned all of the basic identities in greater depth. This also helped me learn how to work through this kind of identity and do it again in the future. |