Functions of angles in a right triangle

Custom Search

Functions and cofunctions

Basic Engineering

Side of Right Triangle by Tangent

FUNCTIONS OF ANGLES IN A RIGHT TRIANGLE

For an acute angle in a right triangle, the length of the side opposite the angle corresponds to y and the length of the side adjacent to the angle corresponds to x, while the length of the hypotenuse corresponds to r. Therefore, the functions of an acute angle in a right triangle can be stated as follows:

If you consider a 90 angle with respect to the "circle of unit radius" diagram, you will realize that for a 90 angle, x = 0, y = 1, and r (as always) equals 1. Since sine = y/r, it follows that the sine of 90 = 1. Since cosine = X/r, it follows that the cosine of 90 = 0/1, or 0. Since tangent= y/x, it follows that tan 90 = 1/0, or infinity (00). From one standpoint, division by 0 is a mathematical inpossibility, since it is impossible to state how many zeros there are in anything. From this standpoint, tan 90 is simply impossible. From another standpoint it can be said that there arc an "infinite" number of zeros in 1. From that standpoint, tan 90 can be said to be infinity.

In real life, the sides of a right triangle y, x, and r, or side opposite, side adjacent, and hypotenuse, are given other names according to the circumferences. In connection with a pitched roof rafter, for instance, y or side opposite is "total rise," x or side adjacent is "total run," and r or hypotenuse is "rafter length." In connection with a ground slope, y or side opposite is "vertical rise," x or side adjacent is "horizontal distance," and r or hypotenuse is "slope distance."

METHODS OF SOLVING TRIANGLES

To "solve" a triangle means to determine one or more unknown values (such as the length of a side or the size of an angle) from given known values. Here are some of the methods used.

Pythagorean Theorem

When you know the lengths of two sides of a right triangle, or its hypotenuse and one side, you can determine the length of the remaining side, or the length of the hypotenuse, by applying the Pythagorean theorem. The Pythagorean theorem states that the square of the length of the hypotenuse of any right triangle equals the sum of the squares of the lengths of the other two sides.

Figure 1-23 shows a right triangle with acute angles A and B and right angle C. Sides opposite A and B are designated as a and b; the hypotenuse (opposite C) is designated as c. Side a measures 3.00 ft, side b measures 4.00 ft, and the hypotenuse measures 5.00 ft. Any triangle with sides and hypotenuse in the ratio of 3:4:5 is a right triangle.

If C²= a²+ b², it follows that c =

The formulas for solving for either side, given the other side and the hypotenuse; or for the hypotenuse, given the two sides, are

Figure 1-23.-A right triangle.

Acute Angle of Right Triangle by Tangent

One of the angles in a right triangle always measures 900. Because the sum of the three angles in any triangle is always 180, it follows that each of the other two angles in a right triangle must be an acute (less than 90) angle. Also, if you know the size of one of the acute angles, you can determine the size of the other from the formulas A = (90 B) and B = (90 A).

In any right triangle in which you know the lengths of the sides, you can determine the size of either of the acute angles by applying the tangent of the angle. Take angle A in figure 1-23, for example. You know that

Reference to a table of natural tangents shows that an angle with tangent 0.75 measures to the nearest minute, 3652'.