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Ratio Identities

Ratio Identities

Trig identities defining tangent and cotangent in terms of sine and cosine.

 

Ratio Identities

Ratio identity formula: tan x equals sin x divided by cos x     Ratio identity formula: cot x equals cos x divided by sin x

Key Formula

tan⁡θ=sin⁡θcos⁡θcot⁡θ=cos⁡θsin⁡θ\tan\theta = \frac{\sin\theta}{\cos\theta} \qquad \cot\theta = \frac{\cos\theta}{\sin\theta}
Where:
  • θ\theta = Any angle (in degrees or radians) for which the expression is defined
  • sin⁡θ\sin\theta = The sine of the angle θ
  • cos⁡θ\cos\theta = The cosine of the angle θ
  • tan⁡θ\tan\theta = The tangent of the angle θ, undefined when cos θ = 0
  • cot⁡θ\cot\theta = The cotangent of the angle θ, undefined when sin θ = 0

Worked Example

Problem: Given that sin θ = 3/5 and cos θ = 4/5, find tan θ and cot θ using the ratio identities.
Step 1: Write the ratio identity for tangent.
tan⁡θ=sin⁡θcos⁡θ\tan\theta = \frac{\sin\theta}{\cos\theta}
Step 2: Substitute the known values of sin θ and cos θ.
tan⁡θ=3545=35×54=34\tan\theta = \frac{\frac{3}{5}}{\frac{4}{5}} = \frac{3}{5} \times \frac{5}{4} = \frac{3}{4}
Step 3: Write the ratio identity for cotangent.
cot⁡θ=cos⁡θsin⁡θ\cot\theta = \frac{\cos\theta}{\sin\theta}
Step 4: Substitute the known values.
cot⁡θ=4535=45×53=43\cot\theta = \frac{\frac{4}{5}}{\frac{3}{5}} = \frac{4}{5} \times \frac{5}{3} = \frac{4}{3}
Step 5: Notice that tan θ and cot θ are reciprocals of each other, which serves as a quick check.
tan⁡θ⋅cot⁡θ=34⋅43=1✓\tan\theta \cdot \cot\theta = \frac{3}{4} \cdot \frac{4}{3} = 1 \checkmark
Answer: tan θ = 3/4 and cot θ = 4/3.

Another Example

This example shows how ratio identities are used to simplify trigonometric expressions, rather than just evaluating numerical values. This is the most common way ratio identities appear on tests.

Problem: Simplify the expression (sin θ · cot θ) using a ratio identity.
Step 1: Replace cot θ with its ratio identity definition.
sin⁡θ⋅cot⁡θ=sin⁡θ⋅cos⁡θsin⁡θ\sin\theta \cdot \cot\theta = \sin\theta \cdot \frac{\cos\theta}{\sin\theta}
Step 2: Cancel the common factor of sin θ in the numerator and denominator.
=sin⁡θ⋅cos⁡θsin⁡θ=cos⁡θ= \frac{\sin\theta \cdot \cos\theta}{\sin\theta} = \cos\theta
Step 3: State the simplified result.
sin⁡θ⋅cot⁡θ=cos⁡θ\sin\theta \cdot \cot\theta = \cos\theta
Answer: sin θ · cot θ simplifies to cos θ.

Frequently Asked Questions

Why are they called ratio identities?
They are called ratio identities because tangent and cotangent are each defined as a ratio (a fraction) of the two fundamental trig functions, sine and cosine. The word 'identity' means these equations are true for every angle where the expression is defined, not just for specific values.
What is the difference between ratio identities and reciprocal identities?
Ratio identities express tan θ and cot θ as fractions involving sin θ and cos θ. Reciprocal identities, on the other hand, pair each trig function with its reciprocal: csc θ = 1/sin θ, sec θ = 1/cos θ, and cot θ = 1/tan θ. While cot θ appears in both sets, the ratio identity writes it as cos θ/sin θ, whereas the reciprocal identity writes it as 1/tan θ.
When are the ratio identities undefined?
The tangent ratio identity is undefined whenever cos θ = 0, which occurs at θ = 90°, 270°, or more generally θ = 90° + 180°n for any integer n. The cotangent ratio identity is undefined whenever sin θ = 0, which occurs at θ = 0°, 180°, 360°, or more generally θ = 180°n.

Ratio Identities vs. Reciprocal Identities

Ratio IdentitiesReciprocal Identities
DefinitionExpress tan and cot as ratios of sin and cosExpress csc, sec, cot as reciprocals of sin, cos, tan
Key formulastan θ = sin θ / cos θ, cot θ = cos θ / sin θcsc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ
Number of identities2 identities3 identities
Primary useRewrite tan or cot in terms of sin and cos to simplify expressionsReplace a trig function with 1 over its reciprocal function

Why It Matters

Ratio identities are among the first trig identities you learn, and they appear constantly when simplifying expressions, verifying other identities, and solving trig equations. In calculus, rewriting tan θ and cot θ in terms of sin θ and cos θ is a standard technique for integration and differentiation. Mastering these two simple identities makes every other category of trig identity easier to work with.

Common Mistakes

Mistake: Mixing up the numerator and denominator — writing tan θ = cos θ / sin θ instead of sin θ / cos θ.
Correction: Remember that tangent = sine over cosine. A helpful mnemonic: in alphabetical order, cosine comes before sine, and cosine goes on the bottom (denominator) of tangent.
Mistake: Forgetting domain restrictions and using tan θ or cot θ at angles where they are undefined.
Correction: Always check that the denominator is not zero. Tan θ is undefined when cos θ = 0 (at odd multiples of 90°), and cot θ is undefined when sin θ = 0 (at multiples of 180°).

Related Terms