Trigonometric Functions of a Single Angle

    The trigonometric functions for any angle are defined and it is shown how to evaluate the trigonometric functions of an angle whose terminal side passes through a given point. The relationships between the trigonometric functions of an angle and the trigonometric functions of the negative of the angle are given. Reference angles are then defined and it is illustrated how to use the reference angle to evaluate the six trigonometric functions.

Definition (Trigonometric Functions) If is a point other than the origin on a circle of radius the radius sweeps out an angle in standard position. The trigonometric functions of are

    If is a point other than the origin on a circle of radius the radius sweeps out an angle in standard position. If is acute, then we can use the definition of the trigonometric functions to write them in terms of the coordinates of as follows:













From these equations you should notice that the definitions of the trigonometric functions of any angle are consistent with the definitions of the trigonometric functions of an acute angle.

Definition (Trigonometric Functions of a Real Number) The value of a trigonometric function at a real number its value at an angle of radians, provided that value exists.

Example (Trigonometric Functions) Given the point we can evaluate the six trigonometric functions of the angle in standard position whose terminal side lies along the line through the origin and the point We find the distance between the point and the origin either using the Pythagorean Theorem or the distance formula, to have Therefore, the six trigonometric functions of this are

    The signs of the trigonometric functions are determined by the signs of the coordinates of the point For example, if is the angle whose terminal side lies on the line extending from the origin to then is negative when and is positive when because Also, is negative when and is positive when because In this manner the sign for all six trigonometric functions of can be determined. In particular, the sine function is positive in the first and second quadrants and is negative in the third and fourth quadrants. The cosine function is positive in the first and fourth quadrants and is negative in the second and third quadrants.

Example (Signs of the Trigonometric Functions) The trigonometric fuctions can be positive or negative depending on which quadrant is in. To summarize this we have:

Proposition (Formulas for Negatives) Using the definitions of the six trigonometric functions we can determine the values for negative angles in terms of values of positive angles and in summary we have:

Example (Trigonometric Functions of Negative Angles) Evaluate the six trigonometric functions of We have

Definition (Reference Angle) If is a point other than the origin on a circle of radius the radius sweeps out an angle in standard position. When a perpendicular is dropped from to the axis an acute angle is determined by the hypotenuse and the axis, which we call the reference angle of

Example (Reference Angle) The reference angle of is The reference angle of is The reference angle of is The reference angle of is because

Proposition (Reference Angle) The trigonometric function of any angle is equal to the same trigonometric function of the reference angle of except for a possible difference of sign. The quadrant in which the terminal side of lies determines the sign of the trigonometric function.

Example (Using Reference Angles) Evaluate the six trigonometric functions of The reference angle of is and lies in quadrant two, so we have

Definition (Periodic Function) A function is periodic if there exists a positive real number such that for every in the domain of The least positive real number if it exists, is the period of

Proposition (Periodic Function) The six trigonometric functions are periodic functions.

Definition (Even Function) A function of a real number is called an even function when for all in the domian of

Definition (Even Function) A function of a real number is called an odd function when for all in the domian of

Proposition (Parity of the Trigonometric Functions) The cosine and secant functions are even functions and the sine, cosecant, tangent and cotangent functions are odd functions.

Proposition (Basic Features of the Trigonometric Functions) The basic features of the sine, cosine, secant, cosecant, tangent, and cotangent functions are summarized in the following graphs and table.




    


    

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