Ratio Identities

High School

Definition

Triganometry identities that define TAN and COT, in terms of SIN and COS. It is a means of simplifying equations by writing two trigonometry functions in terms of basic functions. TAN is a ratio of SIN and COS while COT is the ratio of COS and SIN.

Worked examples

\(\tan\theta = \frac{\sin\theta}{\cos\theta}\)

Tangent is the ratio of sine over cosine.

\(\cot\theta = \frac{\cos\theta}{\sin\theta}\)

Cotangent is the ratio of cosine over sine.

\(\tan(30^\circ) = \frac{\sin(30^\circ)}{\cos(30^\circ)} = \frac{1/2}{\sqrt{3}/2} = \frac{1}{\sqrt{3}}\)

Use the ratio identity to compute tangent from known sine and cosine values.

Common mistakes

\(\tan\theta = \frac{\cos\theta}{\sin\theta}\) → \(\tan\theta = \frac{\sin\theta}{\cos\theta}\) Tangent is sine over cosine, not the other way around. The flipped ratio is cotangent.
\(\cot\theta = \frac{1}{\tan\theta} = \frac{\sin\theta}{\cos\theta}\) → \(\cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta}\) When you flip tangent you must also flip the sine-cosine ratio.
\(\tan\theta = \sin\theta \cdot \cos\theta\) → \(\tan\theta = \frac{\sin\theta}{\cos\theta}\) Ratio identities use division, not multiplication.

Where you'll use it next

You'll use ratio identities to simplify trigonometric equations, prove other identities, rewrite expressions in calculus, and solve problems where tangent or cotangent can be replaced by sine and cosine.

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