Trigonometric Identities: Reciprocal, Pythagorean, and Quotient Identities

Trigonometric Identities and Their Applications

Identities in Mathematics

In mathematics, an identity is an equation that holds true for all values of the variables for which the expressions are defined. For example, the algebraic identity $(x + y)^2 = x^2 + 2xy + y^2$ is true for all real numbers x and y.

Reciprocal Identities

Definition and Explanation

Reciprocal identities relate each trigonometric function to its reciprocal. The reciprocal of a number x is $\frac{1}{x}$, and the reciprocal of a trigonometric function is defined wherever the original function is nonzero.

• sin θ and csc θ are reciprocals.

• cos θ and sec θ are reciprocals.

• tan θ and cot θ are reciprocals.

For all angles θ for which both sides are defined:

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

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

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

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

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

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

Calculator Note: To compute the reciprocal of a trigonometric function, use the $x^{-1}$ key or enter $1/\text{function}$. Do not confuse this with the inverse trigonometric function (e.g., $\sin^{-1}$), which returns the angle whose sine is a given value.

Signs and Ranges of Trigonometric Functions

Signs in Different Quadrants

The sign of a trigonometric function depends on the quadrant in which the terminal side of angle θ lies. Let (x, y) be a point on the terminal side of θ, and r the distance from the origin to (x, y) ($r > 0$):

• $\sin \theta = \frac{y}{r}$

• $\csc \theta = \frac{r}{y}$

• $\cos \theta = \frac{x}{r}$

• $\sec \theta = \frac{r}{x}$

• $\tan \theta = \frac{y}{x}$

• $\cot \theta = \frac{x}{y}$

The signs of these functions in each quadrant are as follows:

Quadrant	x	y	Functions Positive
I	+	+	All
II	-	+	sin, csc
III	-	-	tan, cot
IV	+	-	cos, sec

Mnemonic: "All Students Take Calculus" (or "All Silver Tea Cups") helps remember which functions are positive in each quadrant:

• All (Quadrant I): All functions positive

• Students/Silver (Quadrant II): sin and csc positive

• Take/Tea (Quadrant III): tan and cot positive

• Calculus/Cups (Quadrant IV): cos and sec positive

Ranges of Function Values

• $\sin \theta$ and $\cos \theta$ are always between -1 and 1:

  • $-1 \leq \sin \theta \leq 1$

  • $-1 \leq \cos \theta \leq 1$

• $\tan \theta$ and $\cot \theta$ can take any real value (unbounded).

• $\csc \theta$ and $\sec \theta$ are always $\leq -1$ or $\geq 1$ (never between -1 and 1).

Pythagorean Theorem and Trigonometric Functions

Relationship Between x, y, and r

For a point (x, y) on the terminal side of angle θ, the radius r is given by the Pythagorean Theorem:

• $r^2 = x^2 + y^2$, with $r > 0$

This relationship allows us to find all six trigonometric function values if we know one function value and the quadrant of θ.

Pythagorean Identities

Derivation and Forms

Dividing the Pythagorean Theorem by $r^2$ gives:

• $\left(\frac{x}{r}\right)^2 + \left(\frac{y}{r}\right)^2 = 1$

• Which is $\cos^2 \theta + \sin^2 \theta = 1$

Other Pythagorean identities can be derived by dividing by $\sin^2 \theta$ or $\cos^2 \theta$:

• $\tan^2 \theta + 1 = \sec^2 \theta$

• $1 + \cot^2 \theta = \csc^2 \theta$

Note: $\sin^2 \theta$ means $(\sin \theta)^2$, not $\sin(\theta^2)$.

Quotient Identities

Definitions

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

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

These identities are useful for rewriting expressions and solving equations involving trigonometric functions.

Summary Table: Fundamental Trigonometric Identities

Type	Identity
Reciprocal	$\sin \theta = \frac{1}{\csc \theta}$ $\csc \theta = \frac{1}{\sin \theta}$ $\cos \theta = \frac{1}{\sec \theta}$ $\sec \theta = \frac{1}{\cos \theta}$ $\tan \theta = \frac{1}{\cot \theta}$ $\cot \theta = \frac{1}{\tan \theta}$
Pythagorean	$\sin^2 \theta + \cos^2 \theta = 1$ $\tan^2 \theta + 1 = \sec^2 \theta$ $1 + \cot^2 \theta = \csc^2 \theta$
Quotient	$\tan \theta = \frac{\sin \theta}{\cos \theta}$ $\cot \theta = \frac{\cos \theta}{\sin \theta}$

Example

Example: If $\sin \theta = \frac{3}{5}$ and θ is in Quadrant II, find $\cos \theta$ and $\tan \theta$.

• Since $\sin \theta = \frac{y}{r} = \frac{3}{5}$, let $y = 3$, $r = 5$.

• Find x using $r^2 = x^2 + y^2$: $5^2 = x^2 + 3^2 \implies x^2 = 25 - 9 = 16 \implies x = -4$ (negative in Quadrant II).

• $\cos \theta = \frac{x}{r} = \frac{-4}{5}$

• $\tan \theta = \frac{y}{x} = \frac{3}{-4} = -\frac{3}{4}$

Additional info: The above example demonstrates how to use the Pythagorean Theorem and quadrant information to determine all trigonometric function values from one known value.