Consider the
right triangle pictured below:
Figure 2.1: A right triangle with vertex A at the origin and angle A in standard
position
Using the lengths of the sides of right triangles such as the one above, the
trigonometric functions can be defined in the following way:
In order to solve a right triangle, you must first figure out which angle is the
right angle. Knowing the right angle will also tell you which side is the
hypotenuse, since the
hypotenuse will always stand
opposite the right angle.
In this text, for the sake of consistency, in all triangles we will designate
angle
C as the right angle, and side
c and the hypotenuse. To finish solving
a right triangle, you then must either know the lengths of two sides, or the
length of one side and the measure of one
acute angle. Given either of these
two situations, a triangle can be solved. Any further information about a
triangle may be helpful, but it is not necessary.
There are four basic techniques to use in solving triangles.
- Using the Pythagorean Theorem, once two
sides are known, the third side can be calculated.
- Using the fact that the acute angles of a right triangle are complementary,
once one acute angle is known, the other can be calculated.
- Using the definitions of the trigonometric functions, any two parts of a
triangle can be related in an equation to equal a third part.
- Using the definitions of the inverse trigonometric
functions, any two sides of a
triangle can be related in an equation to equal the inverse function of an
unknown acute angle.
The last two techniques are the most difficult to understand. Some examples
will help clear them up.
Using technique #3, given
a = 4 and
B = 22o,
c = a sec(B) = 
. In this example, we will use trigonometric function
definitions to calculate an unknown part, side
c. A calculator (or a very good
memory) is necessary to evaluate certain function values, like
sec(B) and
cos(B) in this example. In this way trigonometric functions can be used to
calculate unknown parts of triangles.
Using technique #4, given
a = 3 and
b = 4,
= arctan(A) = arccot(B). Here the inverse functions Arctangent and Arccotangent are used to
calculate the measures of either unknown acute angle in a particular triangle.
Again, a calculator is necessary to do the final calculation. There are
numerous ways to relate any two parts of a triangle in a trigonometric equation
to find a third unknown part.
