Trigonometric Functions of Any Angle

Definitions of Trigonometric Functions of Any Angle

Trigonometric functions can be defined for any angle, not just acute angles in right triangles. For an angle θ in standard position, with its terminal side passing through the point P = (x, y) and distance r = \sqrt{x^2 + y^2} from the origin, the six trigonometric functions are defined as:

• Sine:

• Cosine:

• Tangent: ,   x ≠ 0

• Cosecant: ,   y ≠ 0

• Secant: ,   x ≠ 0

• Cotangent: ,   y ≠ 0

Example: If the terminal side of θ passes through (3, 4), then , so , , , etc.

Trigonometric Functions of Quadrantal Angles

Quadrantal angles are angles whose terminal sides lie on the x- or y-axis (e.g., 0°, 90°, 180°, 270°, or their radian equivalents). The trigonometric functions for these angles can be evaluated by considering the coordinates of the point where the terminal side intersects the unit circle.

• At θ = 0° (or 0 radians), the point is (1, 0), so , , .

• At θ = 90° (or radians), the point is (0, 1), so , , is undefined.

• Similar reasoning applies for 180°, 270°, and 360°.

The Signs of the Trigonometric Functions

The sign of a trigonometric function depends on the quadrant in which the terminal side of the angle lies:

• Quadrant I: All trigonometric functions are positive.

• Quadrant II: Only sine and cosecant are positive.

• Quadrant III: Only tangent and cotangent are positive.

• Quadrant IV: Only cosine and secant are positive.

Example: If θ is in Quadrant III, then is negative, is negative, but is positive.

Reference Angles

A reference angle is the acute angle formed by the terminal side of θ and the x-axis. Reference angles are always positive and less than 90° (or radians). They are used to evaluate trigonometric functions for any angle by relating them to the corresponding acute angle in the first quadrant.

• For θ in Quadrant I: Reference angle = θ

• For θ in Quadrant II: Reference angle = 180° – θ (or )

• For θ in Quadrant III: Reference angle = θ – 180° (or )

• For θ in Quadrant IV: Reference angle = 360° – θ (or )

Example: The reference angle for 210° is 210° – 180° = 30°.

Using Reference Angles to Evaluate Trigonometric Functions

To evaluate a trigonometric function for any angle:

1. Find the reference angle for θ.

2. Evaluate the trigonometric function for the reference angle.

3. Determine the correct sign based on the quadrant in which θ lies.

Example: To find :

• Reference angle = 180° – 150° = 30°

• 150° is in Quadrant II, where sine is positive, so

Procedure Summary:

• Identify the quadrant of θ.

• Find the reference angle.

• Evaluate the function for the reference angle.

• Apply the correct sign.

Additional info: For angles greater than 360° (or ) or less than –360° (or –$2\pi$), subtract or add multiples of 360° (or $2\pi$) to find a coterminal angle between 0° and 360° (or 0 and $2\pi$), then proceed as above.