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Trigonometry For DummiesFrom Trigonometry For Dummies, 2nd Edition by Mary Jane SterlingTrigonometry is the study of triangles, which contain angles, of course. Get to know some special rules forangles and various other important functions, definitions, and translations. Sines and cosines are two trigfunctions that factor heavily into any study of trigonometry; they have their own formulas and rules that you’llwant to understand if you plan to study trig for very long.

Formulas to Help You in TrigonometryMany of the formulas used in trigonometry are also found in algebra andanalytic geometry. But trigonometry also has some special formulas usuallyfound just in those discussions. A formula provides you a rule or equation thatyou can count on to work, every single time. A formula gives a relationshipbetween particular quantities and units. The main trick to using formulas is toknow what the different letters represent. In the formulas given here, you have:r (radius); d (diameter or distance); b (base or measure of a side); h (height); a,b, c (measures of sides); x, y (coordinates on a graph); m (slope); M (midpoint);h, k (horizontal and vertical distances from the center); θ (angle theta); and s(arc length). The formulas particular to trigonometry have: sin (sine), cos(cosine), and tan (tangent), although only sin is represented here.

Special Right TrianglesEvery right triangle has the property that the sum of the squares of the two legsis equal to the square of the hypotenuse (the longest side). The Pythagorean

theorem is written: a2 + b2 = c2. What’s so special about the two right trianglesshown here is that you have an even more special relationship between themeasures of the sides — one that goes beyond (but still works with) thePythagorean theorem. When you have a 30­60­90 right triangle, the measureof the hypotenuse is always twice the measure of the shortest side, and theother leg is always

or about 1.7 times as big as the shortest side. With the isosceles right triangle,the two legs measure the same, and the hypotenuse is always

or about 1.4 times as long as those two legs.

Right Triangle Definitions for Trigonometry FunctionsThe basic trig functions can be defined with ratios created by dividing thelengths of the sides of a right triangle in a specific order. The label hypotenusealways remains the same — it’s the longest side. But the designations ofopposite and adjacent can change — depending on which angle you’rereferring to at the time. The opposite side is always that side that doesn’t helpmake up the angle, and the adjacent side is always one of the sides of theangle.

Coordinate Definitions for Trigonometry FunctionsThe trig functions can be defined using the measures of the sides of a righttriangle. But they also have very useful definitions using the coordinates ofpoints on a graph. First, let let the vertex of an angle be at the origin — thepoint (0,0) — and let the initial side of that angle lie along the positive x­axisand the terminal side be a rotation in a counterclockwise motion. Then, whenthe point (x,y) lies on a circle that’s intersected by that terminal side, the trigfunctions are defined with the following ratios, where r is the radius of thecircle.

Signs of Trigonometry Functions in QuadrantsAn angle is in standard position when its vertex is at the origin, its initial side ison the positive x­axis, and the terminal side rotates counterclockwise from theinitial side. The position of the terminal side determines the sign of the varioustrig functions of that angle. The following shows you which functions arepositive — and you can assume that the other functions are negative in thatquadrant.

Degree/Radian Equivalences for Selected AnglesAs you study trigonometry, you'll find occasions when you need to changedegrees to radians, or vice versa. A formula for changing from degrees toradians or radians to degrees is:

The formula works for any angle, but the most commonly used angles andtheir equivalences are shown below.

Laws of Sines and CosinesThe laws of sines and cosines give you relationships between the lengths ofthe sides and the trig functions of the angles. These laws are used when youdon’t have a right triangle — they work in any triangle. You determine whichlaw to use based on what information you have. In general, the side a liesopposite angle A, the side b is opposite angle B, and side c is opposite angleC.

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Exact Trigonometry Functions for Selected AcuteAnglesUsing the lengths of the sides of the two special right triangles — the 30­60­90right triangle and the 45­45­90 right triangle — the following exact values fortrig functions are found. Using these values in conjunction with referenceangles and signs of the functions in the different quadrants, you can determinethe exact values of the multiples of these angles.

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