Fundamental Trig Identities: Reciprocal & Quotient Rules

Foundational

Master the six reciprocal identities and two quotient identities. These are the foundational building blocks for simplifying expressions and proving more complex trigonometric equations. Knowing these by heart is the first step to success in trigonometry.

Fundamental Trig Identities: Reciprocal & Quotient Rules

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What is the reciprocal identity for cosecant, $\csc(\theta)$?

The reciprocal of sine is cosecant. $$ \csc(\theta) = \frac{1}{\sin(\theta)} $$

What is the reciprocal identity for secant, $\sec(\theta)$?

The reciprocal of cosine is secant. $$ \sec(\theta) = \frac{1}{\cos(\theta)} $$

What is the reciprocal identity for cotangent, $\cot(\theta)$?

The reciprocal of tangent is cotangent. $$ \cot(\theta) = \frac{1}{\tan(\theta)} $$

Express sine, $\sin(\theta)$, using its reciprocal function.

The reciprocal of cosecant is sine. $$ \sin(\theta) = \frac{1}{\csc(\theta)} $$

Express cosine, $\cos(\theta)$, using its reciprocal function.

The reciprocal of secant is cosine. $$ \cos(\theta) = \frac{1}{\sec(\theta)} $$

Express tangent, $\tan(\theta)$, using its reciprocal function.

The reciprocal of cotangent is tangent. $$ \tan(\theta) = \frac{1}{\cot(\theta)} $$

What is the quotient identity for tangent, $\tan(\theta)$, using sine and cosine?

Tangent is the ratio of sine to cosine. $$ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} $$

What is the quotient identity for cotangent, $\cot(\theta)$, using sine and cosine?

Cotangent is the ratio of cosine to sine. $$ \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} $$

Fundamental Trig Identities: Reciprocal & Quotient Rules

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