Quotient identities

Definition

Quotient identities are trigonometric identities that express tangent and cotangent functions as the quotient of sine and cosine functions. Specifically, $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$ and $\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\cos(\theta)}{\sin(\theta)}$.

5 Must Know Facts For Your Next Test

1. $\tan(\theta)$ is defined as $\frac{\sin(\theta)}{\cos(\theta)}$.
2. $\cot(\theta)$ is defined as $\frac{1}{\tan(\theta)}$ or equivalently $ \frac{\cos(\theta)}{\sin(\theta)}$.
3. Quotient identities are useful for simplifying trigonometric expressions and verifying other trigonometric identities.
4. The quotient identities can be derived from the definitions of sine, cosine, tangent, and cotangent.
5. In contexts where either $\sin(\theta)$ or $\cos(\theta)$ is zero, the quotient identities may lead to undefined expressions.

Review Questions

• What is the quotient identity for $\tan(\theta)$?
• How can you express $\cot(\theta)$ using sine and cosine?
• Why are quotient identities important when simplifying trigonometric expressions?

Related terms

Pythagorean Identities: Identities expressing the fundamental relationship between sine, cosine, and their squares: $ \sin^2(\theta) + \cos^2(\theta) = 1$, among others.

Reciprocal Identities: Identities that express trigonometric functions as reciprocals of each other: $ \csc(\theta) = \frac{1}{\sin(\theta)}, \sec(\eta} = \frac{1}{\cos(\eta)}, \cot(\eta} = \frac{1}{\tan(\eta}$.

Trigonometric Functions: $\sin$, $\cos$, $\tan$, $\csc$, $\sec$, and $\cot$ functions based on angles in a right triangle or unit circle.