Tangent Identity
Defines tangent in terms of sine and cosine.
Formula
Variables
Example
tan 45° = sin45°/cos45° = (√2/2)/(√2/2) = 1.
How it works
The tangent of an angle is the ratio of the sine of that angle to its cosine. This comes straight from the right-triangle definitions: sin θ is opposite/hypotenuse and cos θ is adjacent/hypotenuse, so dividing them cancels the hypotenuse and leaves opposite/adjacent, which is tangent. It is also useful on the unit circle, where tan θ can be viewed as the y-coordinate divided by the x-coordinate, again giving sin θ / cos θ.
When to use it
Use this identity when you already know sine and cosine and need tangent without calculating a separate trig value. It is especially helpful in trigonometric simplification, solving equations, and checking whether an angle is in a quadrant where tangent should be positive or negative.
Worked example, step by step
Find tan 30° using the identity. sin 30° = 1/2 cos 30° = √3/2 tan 30° = sin 30° / cos 30° tan 30° = (1/2) / (√3/2) tan 30° = (1/2) × (2/√3) tan 30° = 1/√3 tan 30° = √3/3
Common mistakes to avoid
- Reversing the fraction and writing cos θ / sin θ instead of sin θ / cos θ
- Using the identity when cos θ = 0, which makes tangent undefined
- Forgetting to simplify the fraction correctly after substituting sine and cosine values
Did You Know?
Tangent is sine over cosine — which is why it blows up to infinity wherever cosine hits zero.
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