Signs of Trigonometric Functions
Signs of Trigonometric Functions in Each Quadrant
We know that cos α = x/r. Since r > 0, the sign of the cosine function depends only on the sign of x. Therefore, the cosine is positive in the 1st and 4th quadrants, and negative in the 2nd and 3rd quadrants.
Consider the sine function
It is clear that the reciprocal functions
The signs of tangent and cotangent depend on the signs of sine and cosine. The tangent and cotangent are positive when
We can summarize this information in the following table:
Evaluating Trigonometric Functions
To find a trigonometric function of any angle, it is convenient to use the concept of reference angle. This involves the following steps:
- Determine the reference angle for the given angle in standard position. The reference angle is always acute.
- Calculate the trigonometric function value for the reference angle.
- Determine the sign of the trigonometric function depending on the quadrant in which the terminal side of the given angle lies.
Example
Calculate the value of
Solution.
We denote
The cosine of the reference angle can be found from the table:
The initial angle
Pythagorean Trigonometric Identities for Any Angle
We already know that, for any acute angle, the following identities are valid:
It turns out that these identities remain valid for any angle
Consider, for example, the first identity. Let
It follows from here that
which means
If we know one of the trigonometric functions of an angle and the quadrant in which the angle lies, we can determine all other trigonometric functions of this angle. This can be done using the identities listed above and definitions of trigonometric functions. Note that if a trigonometric expression contains a quotient, it is valid for only those angles at which the denominator is not zero.