Appendix A

Trigonometric Functions

Trigonometry is a very valuable form of mathematics for anyone
who studies electricity/electronics. Trigonometry deals with angles and
triangles, particularly the right triangle, which has one angle of 90°. An
electronic example of a right triangle is shown in Figure A-1. This example
illustrates how resistance, reactance, and impedance are related in AC
circuits. We know that resistance (R) and reactance (X) are 90° apart, so
their angle of intersection forms a right angle. We can use the law of right
triangles, known as the Pythagorean theorem, to solve for the value of any
side. This theorem states that in any right triangle, the square of the hypotenuse
is equal to the sum of the squares of the other two sides. With reference to Figure
A-1, we can express the Pythagorean theorem mathematically as:

Z^2 = R^2 + X^2

or,

Z = R^2 + X^2

By using trigonometric relationships, we can solve problems dealing
with phase angles, power factor, and reactive power in AC circuits. The
three most used trigonometric functions are the sine, the cosine, and the
tangent. These functions show the ratios of the sides of the triangle, which
determine the size of the angles. Figure A-2 illustrates how these ratios are
expressed mathematically, and their values can be found.
Trigonometric ratios hold true for angles of any size; however,
we use angles in the first quadrant of a standard graph (0° to 90°) as a
reference, and in order to solve for angles greater than 90° (second-, third-
, and fourth-quadrant angles), we can convert them to first-quadrant
angles (see Figure A-3). All first-quadrant angles have positive functions,
while angles in the second, third, and fourth quadrants have two negative
functions and one positive function.

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