Tangent Identity
Tangent equals sine divided by cosine.
Formula
Variables
How it works
The tangent of an angle is defined as the ratio of the side opposite the angle to the side adjacent to it in a right triangle. Because sine is opposite divided by hypotenuse and cosine is adjacent divided by hypotenuse, dividing sine by cosine cancels the hypotenuse and leaves opposite divided by adjacent. That is why tan = sin / cos, as long as cos is not zero.
When to use it
Use this identity when you know sine and cosine and want tangent without switching to a triangle or calculator directly. It is especially useful in trig simplification, verifying identities, and finding slopes or angles in geometry and physics problems.
Worked example, step by step
Find tan(30 ). Step 1: Use the identity tan = sin / cos . Step 2: Substitute the exact values: sin(30 ) = 1/2 and cos(30 ) = 3/2. Step 3: Compute the ratio: tan(30 ) = (1/2) / (3/2). Step 4: Multiply by the reciprocal: tan(30 ) = (1/2) (2/3) = 1/3. Final answer: tan(30 ) = 1/3.
Common mistakes to avoid
- Forgetting that the angle must be in a range where the tangent is defined, since cos cannot be 0
- Reversing the ratio and writing cos/sin instead of sin/cos
- Using decimal approximations too early and losing exact values like 1/3
Did You Know?
That tangent equals sine over cosine means the tangent is undefined wherever cosine hits zero, at 90° and 270°.
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More in Trigonometry
View allSine Ratio (SOH)
BasicRatio of the side opposite an angle to the hypotenuse.
Cosine Ratio (CAH)
BasicRatio of the side adjacent to an angle to the hypotenuse.
Tangent Ratio (TOA)
BasicRatio of the opposite side to the adjacent side.
Cosecant
IntermediateReciprocal of sine.
Secant
IntermediateReciprocal of cosine.
Cotangent
IntermediateReciprocal of tangent.