![]() This makes it obvious that the shortest side (length of 1) is opposite the smallest angle (30°). When you sketch a 30°-60°-90° triangle, exaggerate the fact that it’s wider than it is tall (or taller than wide if you tip it up). Remember that 2 is more than ( equals 2, so be must be less than 2) and that the hypotenuse is always the longest side of a right triangle. It has legs of lengths 1 and (about 1.73), and a 2-unit long hypotenuse.ĭon’t make the common error of switching the 2 with the in a 30°-60°-90° triangle. ![]() The 30°-60°-90° in Figure 6-2 is half of a 2-by-2-by-2 equilateral triangle. ![]() When you apply the SohCahToa trig functions and their reciprocals to the 45° angle in the 45°-45°-90° triangle, you get the following trig values:Įvery 30°-60°-90° triangle is half of an equilateral triangle cut straight down the middle along its altitude. The Pythagorean theorem: For any right triangle,, where a and b are the lengths of the triangle’s legs (the sides touching the right angle) and c is the length of its hypotenuse. The Pythagorean theorem gives you the length of its hypotenuse,, or about 1.41. The 45°-45°-90° triangle in Figure 6-2 is half of a 1-by-1 square. The other three trig functions are reciprocals of these: Cosecant (csc) is the reciprocal of sine, secant (sec) is the reciprocal of cosine, and cotangent (cot) is the reciprocal of tangent.īecause so many garden variety calculus problems involve 30°, 45°, and 60° angles, it’s a good idea to memorize the two right triangles in Figure 6-2.įIGURE 6-2: The 45°-45°-90° triangle and the 30°-60°-90° triangle.Įvery 45°-45°-90° triangle is the shape of a square cut in half along its diagonal. (To remember how to spell SohCahToa, note its pronunciation and the fact that it contains three groups of three letters each.) For any angle , SohCahToa uses the initial letters of sine, cosine, and tangent, and the initial letters of hypotenuse, opposite, and adjacent to help you remember the following definitions. SohCahToa is a meaningless mnemonic device that helps you remember the definitions of the sine, cosine, and tangent functions. The side that’s 3 units long in this right triangle is referred to as the opposite side because it’s on the opposite side of the triangle from angle x, and the side of length 4 is called the adjacent side because it’s adjacent to, or touching, angle x.įIGURE 6-1: Sitting around the campfire, studying a right triangle. The longest side of this right triangle (or any right triangle), the diagonal side, is called the hypotenuse. ![]() The three main trig functions (sine, cosine, and tangent) and their reciprocals (cosecant, secant, and cotangent) all tell you something about the lengths of the sides of a right triangle that contains a given acute angle - like angle x in Figure 6-1. The study of trig begins with the right triangle. So, if your trig is rusty - I’m shocked - review these trig basics, or else! Many calculus problems involve trigonometry, and the calculus itself is enough of a challenge without having to relearn trig at the same time. Warming Up with Calculus Prerequisites Chapter 6. Calculus For Dummies, 2nd Edition (2014) Part II. ![]()
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