Introducing the Trigonometric Functions

    Similar triangles is the geometric idea that is used to define the trgionometric functions as ratios of sides of right triangles. Given an angle there are six possibilites for a function to be defined. The trigonometric functions sine, cosine, tangent, cotangent, secant, and cosecant of an acute angle are defined and detailed with examples and comments. Mnemonic devices are given for the definitions and for the functional values for the special angles of 30, 45, and 60 degrees. The reciprocal properties explain that three of the six trigonometric functions (sine, cosine, and tangent) are just reciprocals of three other trigonometric functions (cosecant, secant, and cotangent). Thus, if the values of some of the trigonometric functions are known the values of the others might be easily found. Finally, the Pythagorean Identities allows us to find the values of the six trigonometric functions given only one of the values.

Definition (Introducing the Trigonometric Functions) Given an acute angle in a right triangle as shown, we can define the six trigonometric functions sine, cosine, tangent, cosecant, secant and cotangent as follows:



where is the side opposite to angle and is the side adjacent angle and is the hypotenuse.

    As an aid in using the trigonometric functions we summarize their abbreviations and definitions:

Example (Introducing the Trigonometric Functions) Trigonometric Functions. Consider the following right triangle.


(a) Given and find the six trigonometric functions of We have







(b) Given and find the six trigonometric functions of We have

Example (Introducing the Trigonometric Functions) Special Angles. Evaluate the six trigonometric functions for and

For any triangle:


and for a triangle:


We can summarize in the following table:



and as a mnemonic device we have

Example (Introducing the Trigonometric Functions) Applying a Trigonometric Function. Find if and in the following right triangle.



    Solution. We have and so Thus, and since we are given we have Thus, and so

Proposition (Introducing the Trigonometric Functions) Reciprocal Properties. Given an acute angle in a right triangle the following relationships follow from their definitions:    

Proposition (Introducing the Trigonometric Functions) Pythagorean Identities. If is an acute angle in a right triangle,  then



    Proof. Since the value of and are independent of the size of the triangle with we choose the following right triangle with angle



Then and Using the Pythagorean Theorem we have and upon substitution it follows Dividing both sides of   by yields Finally, dividing both sides of   by yields   

Example (Introducing the Trigonometric Functions) Finding Values. Given a value for one of the six trigonometric functions in a right triangle, we can find the functional value for the other 5 trigonometric functions by using the reciprocal properties and the Pythagorean Identity. For example, suppose We can use to find as follows









Now we have and in the following right triangle:



Therefore, and Using we can find as follows and so Therefore, Finally,

Example (Introducing the Trigonometric Functions) Evaluating a Trigonometric Function. A scientific calculator can be used to evaluate the six trigonometric functions by doing the  following. For example, to compute use the degree mode and then enter You should get To compute use the radians mode and then enter You should get To evaluate you can use and you should get

Example (Introducing the Trigonometric Functions) Solve a Right Triangle. Solve the right triangle given that and for



We have and since it follows that Therefore, Finally, the Pythagorean Theorem yields and so Finally to solve the triangle we need to note that

Proposition (Introducing the Trigonometric Functions) Cofunction Theorem. If two angles are complementary, then any trigonometric function of one of them is equal to its cofunction of the other; and conversely, if any two cofunctions are equal to each other then the angles are complementary.

    Proof. Consider the right triangle



Given two acute angles and The six trigonometric functions are defined if and if and are complementary; and by their definitions and

Example (Introducing the Trigonometric Functions) Solving Equations. The Cofunction Theorem can be used to solve some trigonometric equations.

(a) Solve

    Solution. We have and so which give the solution of

(b) Solve

    Solution. We have

Example (Introducing the Trigonometric Functions) Using Trigonometric Functions. Determine when to use each of the trigonometric function.

(a) Find the values of the six trigonometric functions of given and for



    Solution.   We have and so

  We have and so
  
  We have and so


(b) Find the length of side if and

     Solution.



  We have and so

(c) To four decimal places find and

    Solution.  We have

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Cite this as:
Introducing The Trigonometric Functions
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
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