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Techniques for Solving

Consider the right triangle pictured below:

Figure %: 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:


sin(A) = =    

cos(A) = =    

tan(A) = =    

csc(A) = =    

sec(A) = =    

cot(A) = =    

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.

  1. Using the Pythagorean Theorem, once two sides are known, the third side can be calculated.
  2. Using the fact that the acute angles of a right triangle are complementary, once one acute angle is known, the other can be calculated.
  3. Using the definitions of the trigonometric functions, any two parts of a triangle can be related in an equation to equal a third part.
  4. 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 = 22 o , 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.

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