Reciprocal Identities

Definition and Application

Reciprocal identities express each trigonometric function in terms of its reciprocal. These identities are fundamental in simplifying expressions and solving equations involving trigonometric functions.

• Cosecant:

• Secant:

• Cotangent:

Example:

• If , then

• If , then

• If , then

• If , then

Signs and Ranges of Function Values

Quadrant Analysis and Function Signs

The sign of a trigonometric function depends on the quadrant in which the angle lies. The mnemonic "All Students Take Calculus" helps remember which functions are positive in each quadrant:

• Quadrant I: All functions are positive.

• Quadrant II: Sine and cosecant are positive.

• Quadrant III: Tangent and cotangent are positive.

• Quadrant IV: Cosine and secant are positive.

Example: Determine the signs for angles:

• 54º: Quadrant I, all positive.

• 260º: Quadrant III, tangent and cotangent positive.

• -60º: Quadrant IV, cosine and secant positive.

Example: For , , must be in Quadrant III.

Example: Possible or impossible statements:

• : Possible.

• : Impossible (cosine range is [-1, 1]).

• : Impossible (cosecant is undefined for ).

Example: If is in Quadrant III and , find the other functions using reciprocal and Pythagorean identities.

Ratio (Quotient) Identities

Definition and Usage

Ratio identities relate tangent and cotangent to sine and cosine:

These identities are useful for converting between trigonometric functions and simplifying expressions.

Pythagorean Identities

Derivation and Equivalent Forms

Pythagorean identities are derived from the equation of a circle and relate the squares of trigonometric functions:

These identities are valid for all angles for which the function values are defined.

Example: Find and given and :

• Use to solve for .

• Calculate .

Practice: Find and given and is in Quadrant II.

Summary Table: Signs of Trigonometric Functions by Quadrant

Classification Table