Trigonometric Function Identities and Their Properties

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: $\csc \theta = \frac{1}{\sin \theta}$

• Secant: $\sec \theta = \frac{1}{\cos \theta}$

• Cotangent: $\cot \theta = \frac{1}{\tan \theta}$

Example:

• If $\sin \theta = -\frac{2}{3}$, then $\csc \theta = -\frac{3}{2}$

• If $\cos \theta = \frac{3}{5}$, then $\sec \theta = \frac{5}{3}$

• If $\tan \theta = \frac{1}{8}$, then $\cot \theta = 8$

• If $\cos \theta = -\frac{2}{\sqrt{20}}$, then $\sec \theta = -\frac{\sqrt{20}}{2}$

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 $\tan \theta > 0$, $\csc \theta < 0$, $\theta$ must be in Quadrant III.

Example: Possible or impossible statements:

• $\cot \theta = -0.999$: Possible.

• $\cos \theta = -1.7$: Impossible (cosine range is [-1, 1]).

• $\csc \theta = 0$: Impossible (cosecant is undefined for $\sin \theta = 0$).

Example: If $\theta$ is in Quadrant III and $\tan \theta = \frac{8}{5}$, 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:

• $\tan \theta = \frac{\sin \theta}{\cos \theta}$

• $\cot \theta = \frac{\cos \theta}{\sin \theta}$

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 $x^2 + y^2 = r^2$ and relate the squares of trigonometric functions:

• $\sin^2 \theta + \cos^2 \theta = 1$

• $1 + \tan^2 \theta = \sec^2 \theta$

• $1 + \cot^2 \theta = \csc^2 \theta$

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

Example: Find $\cos \theta$ and $\tan \theta$ given $\sin \theta = -\frac{\sqrt{2}}{3}$ and $\cos \theta > 0$:

• Use $\sin^2 \theta + \cos^2 \theta = 1$ to solve for $\cos \theta$.

• Calculate $\tan \theta = \frac{\sin \theta}{\cos \theta}$.

Practice: Find $\sin \theta$ and $\cos \theta$ given $\cot \theta = -\frac{7}{24}$ and $\theta$ is in Quadrant II.

Summary Table: Signs of Trigonometric Functions by Quadrant

Classification Table

Quadrant	sin θ	csc θ	cos θ	sec θ	tan θ	cot θ
I	+	+	+	+	+	+
II	+	+	-	-	-	-
III	-	-	-	-	+	+
IV	-	-	+	+	-	-