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Trigonometry

The word "Trigonometry" is derived from two Greek words meaning measurement or solution of triangles. Trigonometry is a branch of mathematics that deals with the ratio between the sides of a right triangle and its angles. Trigonometry is used in surveying to determine heights and distances, in navigation to determine location and distances, and in fields like nondestructive testing for determining things such as the angle for reflection or refraction of an ultrasound wave.

There are three principle functions in trigonometry: sine A, cosine A, and tangent A where A is an angle. These are typically abbreviated for use in algebra to: sinA, cosA, and tanA. These terms are defined in terms of a right triangle. SinA is equal to the side opposite of the angle A (side a) divided by the hypotenuse of the triangle (side c). From the right triangle below, it can be seen that the value of angle A is directly linked to the ratio of side a and side c. In other words, if the length of side a is changed (and side c is not changed by the same amount), then angle A must change. When the division of the length of side a by side c is performed, the resulting value is directly related to angle A. The relationship between the decimal value of the ratio of the side and the angular value of angle A can be looked up in trigonometry tables or, as is more common these days, programmed into a scientific calculator. The relationship between angle A and the ratio of any two of the three sides can be determined by using either the Sin, Cos or Tan functions. CosA is equal to the side adjacent to the angle A divided by the hypotenuse and tanA is the sine divided by the cosine and is therefore the side opposite the angle divided by the side adjacent.

Angles can be measured in either radians or degrees. 180º = p radians.

Trigonomic Properties

There are several rules to make trigonometry easier. The first rule is the law of sines. This rule is valid for all triangles and is not restricted to right triangles. The law of sines is shown below.

The second rule is the law of cosines. As for the law of sines, this rule is valid for all triangles regardless of the angles. The law of cosines can be seen below.

For the special case of the right triangle, C = 90º and the third term drops out to give the Pythagorean theorem which does not involve either of the other angles.. The Pythagorean theorem is seen below.

Each triangle has six parts, three sides and three angles. If three of these are known including at least one side, the other three can be calculated using the two laws.

Similar Triangles

Similar triangles are triangles that have the exact same angles as each other but not necessarily the same side lengths. There are certain rules that can be determined about similar triangles. If you superimpose one triangle on the other so that one of the corners and two of the sides match, it can be seen that the third sides of each triangle are parallel to each other. Another rule important later in determining geometric unsharpness in radiology is the ratio of sides. The ratio of the two sides a is the same as the ratio of the two sides b and is also the same as the two sides c. This is illustrated below.