TRIGONOMETRY TRIGONOMETRY: SOLUTION OF TRIANGLES Machinery's Handbook, 31st Edition
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Terminology.— A triangle is a polygon with three sides. The sum of the angle measures in any triangle in a plane is 180 degrees. The triangle inequality states that the sum of any two side lengths of a triangle is always greater than the length of the third side. It is not possible to construct a triangle that violates the triangle inequality Triangles are either right or oblique . A right triangle has a right angle, which measures 90 degrees. Oblique triangles do not contain a right angle. As with any polygon (see page 60), parts with equal measure are called congruent . Thus, a triangle with congruent sides is one whose sides have the same measure. A triangle with two congruent sides is called an isosceles triangle; a triangle with all three sides congruent is equilateral and hence equiangular . Each angle measures 180 degrees/3 = 60 degrees. Two positive angles whose measures total 90 degrees are called complementary angles. The two acute angles in any right triangle are complements of each other. Two positive angles whose sum is 180 degrees are called supplementary angles . An isosceles triangle has at least two congruent sides and angles (an equilateral triangle is also isosceles). Angles opposite the congruent sides are congruent angles. An obtuse tri angle has one angle measuring greater than 90 degrees. An acute triangle has all three angles measuring less than 90 degrees; hence, an equilateral (equiangular) triangle is also acute. Degree and Radian Angle Measure.— Two modes of measuring angles are degree mea- sure and radian measure. 1 radian is the measure of a circle’s central angle whose arc is the same length as the radius of the circle. For any size circle, 1 radian is approximately 57.3 degrees. Conversion between degree and radian measure is based on the relation 360° = 2π radians, or 180° = π radians. (π is the ratio of circumference to diameter, C / d , and is approximately 22/7 or 3.1415926. The actual value of π is an irrational number.) Degree measure is converted to its equivalent radian measure by the formula: Degree measure × π /180 = Radian measure For example, 45 ° × π /180 = π /4. Radian measure is converted to degree by the formula: Radian measure × 180/ π = Degree measure For example, π /3 × 180 ° = 60 ° . Radian measure is actually unitless, but it is customary to write “rad” to indicate when radian measure is used. Conversions for the essential degree measures of the circle are shown in the chart on page 95. Trigonometric Ratios of Essential Angles.— An acute angle can be any degree measure between 0° and 90°, but the trigonometric values for base angles (designated as θ in the table below the diagrams) 30°, 45°, and 60° are usually memorized. They are derived from right triangles constructed so that the shortest side has length = 1. The other dimensions follow from the geometric construction of the angles (see Geometric Constructions start- ing on page 66) and the Pythagorean theorem. The triangles and trigonometric values of these angle measures, as well as those with base angles 0° and 90° (envisioned by a horizontal and a vertical line segment, respectively) are given in the diagrams and table below. These five are the essential angles. In the development of the essential angles, the length of the shortest side in each of the triangles is designated as 1, so by construction the other lengths follow. Derivations of the main three functions for the five angle measures is shown. Decimal values of square roots are rounded to three decimal places. In the complete trigonometry tables (Table 2a, Table 2b, and Table 2c), irrational values are carried out to six decimal places, for greater accu- racy. For quick estimates, it is useful to memorize the truncated values in the table below.
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