Precalculus

Trigonometry

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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.

Signs of cosine and secant functions in each quadrant.
Figure 1.

Consider the sine function Its sign is determined by the sign of Hence, the sine is positive in the and quadrants, and negative in the and quadrants.

Signs of sine and cosecant functions in each quadrant.
Figure 2.

It is clear that the reciprocal functions and have the same signs as and respectively.

The signs of tangent and cotangent depend on the signs of sine and cosine. The tangent and cotangent are positive when and are both positive or both negative. This occurs in the and quadrants. In the and quadrants, these functions are negative.

Signs of tangent and cotangent functions in each quadrant.
Figure 3.

We can summarize this information in the following table:

Signs of trigonometric functions
Figure 4.

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:

Example

Calculate the value of

Solution.

We denote The reference angle of is equal to

Reference angle of 7pi/6.
Figure 5.

The cosine of the reference angle can be found from the table:

The initial angle lies in the quadrant in which the cosine function has a negative sign. Therefore,

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 be a point on the terminal side of the angle Then no matter which quadrant the angle belongs to, we can always build a right triangle with the legs and hypotenuse Hence,

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.

See solved problems on Page 2.

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