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Introduction to Trigonometric Ratios Trigonometry is the study of triangles, most commonly, right triangles. Since similar triangles have the same angles and proportional sides, then the following proportion statements are all true: A H A H a h x 90º 90º x 90º B x b B A/H = A/H = a/h B/H = B/H = b/h A/B = A/B = a/b These ratios remain the same for any right triangle with an acute angle of xº regardless of the size of the triangle. So we give them each a name. Sine (xº) Cosine (xº) and Tangent (xº) These are just names of functions that refer to the ratios above: Sine (xº) = A/H Cosine (xº) = B/H Tangent (xº) = A/B This is true even if the triangles are rotated. A is always across from x, B is always next to X, and H is always the longest side (the hypotenuse). A x B A b h H H x a x B 3 x 4 5 sin(x) = 4/5 = .8000 cos(x) = 3/5 = .6000 tan(x) = 4/3 = 1.3333 Trigonometry cont. “soh, cah, toa” sine X = opposite leg hypotenuse cosine X = adjacent leg hypotenuse hypotenuse tangent X = opposite leg = adjacent leg opposite leg angle X adjacent leg If you know the angle measure of “x” and any one side, you can find the other sides using a trig ratio. ? sin(60º) = ?/10 .8660 = ?/10 8.66 = ? 10 60º cos(52º) = 3/? ?*cos(52º) = 3 ? = 3/cos(52º) = 4.87 3 52º ? Try these: 7 ? 8 49º 61º ? First, decide which ratio you need. Then write an equation involving the information you know. On the calculator, push the sin/cos/tan button and then type the angle and hit enter. Be sure you are in degree mode. Put this value into your ratio equation and solve for the unknown side length. Finding The Angle If You Know at Least Two Sides If you know two of the sides and want to find the angle, you need to use the inverse functions (sin-1/cos-1/tan-1). On the calculator, press the 2nd key and then the trig function key needed. Enter the ratio of the two sides and hit enter. Ratio: Angle (use the inverse functions) tanX = 1 then X =tan-1(1) = or 2nd tan (1) on the calculator = 45º sinX = 0.8660 X= sin-1(0.8660) = 60º cosX = 0.7986 X= cos-1(0.7986) = 37º tanX = 5/7 X= tan-1(5/7) = 35.5º So you can use this in a triangle: sin ( x) = 12/13 so: sin-1(12/13) = x = 67.4º 12 3 13 xº xº tan(x) = 3/4 so: tan -1(3/4) = x x = 36.9º 4 Don’t forget you can use the Pythagorean Theorem to find the third side! Try to find the missing values: roof pitch is 7:12 ? 5’ ? ?º