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The Tangent Calculator computes the tangent of any angle in degrees or radians. Tangent is the third primary trigonometric function and is indispensable in applications involving slopes, angles of elevation, and rates of change.
In a right triangle, tangent is the ratio of the opposite side to the adjacent side:
$$\tan\theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{\sin\theta}{\cos\theta}$$
Unlike sine and cosine, tangent is unbounded — it can take any real value from $$-\infty$$ to $$+\infty$$. The tangent function is undefined where $$\cos\theta = 0$$, which occurs at $$\theta = 90°$$ and $$\theta = 270°$$ (and their periodic equivalents). At these points, the function has vertical asymptotes.
Tangent has a period of $$\pi$$ radians (180°), half the period of sine and cosine: $$\tan(\theta + 180°) = \tan\theta$$. It is an odd function: $$\tan(-\theta) = -\tan\theta$$.
The standard exact values are:
$$\tan 0° = 0, \quad \tan 30° = \frac{1}{\sqrt{3}}, \quad \tan 45° = 1, \quad \tan 60° = \sqrt{3}, \quad \tan 90° = \text{undefined}$$
Geometrically, tangent represents the slope of a line making angle $$\theta$$ with the horizontal. A line at 45° has slope $$\tan 45° = 1$$, a line at 60° has slope $$\tan 60° = \sqrt{3} \approx 1.732$$, and a horizontal line has slope $$\tan 0° = 0$$. This direct connection to slope makes tangent the most natural function for angle-related problems in coordinate geometry.
The angle of elevation and angle of depression problems in surveying rely directly on tangent. If you stand at distance $$d$$ from a building and look up at angle $$\theta$$ to see its top, the building height is $$h = d \cdot \tan\theta$$. Similarly, pilots and engineers use tangent to calculate glide paths and road grades.
In calculus, the tangent function has derivative $$\frac{d}{dx}\tan(x) = \sec^2(x) = 1 + \tan^2(x)$$. Its integral is $$-\ln|\cos(x)| + C$$. The Taylor series around zero is: $$\tan\theta = \theta + \frac{\theta^3}{3} + \frac{2\theta^5}{15} + \cdots$$ (converging for $$|\theta| < \pi/2$$).
The inverse tangent function $$\arctan(x)$$ has the convenient property of being defined for all real numbers, returning values in $$(-90°, 90°)$$. It appears frequently in integration formulas and in computing angles from coordinate ratios using $$\theta = \arctan(y/x)$$.
The calculator converts the angle to radians if given in degrees, then computes $$\tan(\theta_{rad})$$. If the cosine of the angle is effectively zero (within $$10^{-10}$$), the result is undefined (NaN). The inverse check applies $$\arctan$$ to return the principal value in $$(-90°, 90°)$$.
Tangent can be any real number. Positive values occur in Quadrants I and III, negative in Quadrants II and IV. An undefined result means the angle is at a vertical asymptote (90°, 270°, etc.). The inverse check returns the principal angle in $$(-90°, 90°)$$.
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tan(45°) = 1 exactly. At this angle, the opposite and adjacent sides of a right triangle are equal. The line makes a 45° angle with a slope of 1.
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tan(120°) = −√3 ≈ −1.7321. In Quadrant II, tangent is negative. The inverse check returns −60° (the principal value), demonstrating that arctan maps to (−90°, 90°).
Since $$\tan\theta = \frac{\sin\theta}{\cos\theta}$$, and $$\cos 90° = 0$$, computing tangent requires division by zero at 90°. Geometrically, a vertical line has an undefined slope. The function approaches $$+\infty$$ from the left and $$-\infty$$ from the right of 90°.
Tangent has a period of $$\pi$$ radians or 180°, which is half the period of sine and cosine. This means $$\tan(\theta + 180°) = \tan\theta$$ for all $$\theta$$. The shorter period reflects the fact that tangent's sign pattern repeats every half-rotation.
The slope of a line making angle $$\theta$$ with the positive x-axis equals $$\tan\theta$$. A 0° angle gives slope 0 (horizontal), 45° gives slope 1, and approaching 90° gives an increasingly steep slope approaching infinity. Negative angles produce negative slopes.
The tangent addition formula is $$\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$$. For subtraction: $$\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$$. The double angle formula $$\tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta}$$ follows directly.
The inverse tangent $$\arctan(x)$$ returns the angle whose tangent is $$x$$. Unlike arcsin and arccos, arctan accepts any real number as input. Its range is $$(-90°, 90°)$$ or $$(-\pi/2, \pi/2)$$. The two-argument function $$\text{atan2}(y, x)$$ is often preferred as it returns the correct angle in all four quadrants.
$$\frac{d}{dx}\tan(x) = \sec^2(x) = 1 + \tan^2(x)$$. This means tangent's rate of change is always at least 1 and increases rapidly near the asymptotes. The integral of tangent is $$-\ln|\cos(x)| + C$$ or equivalently $$\ln|\sec(x)| + C$$.
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The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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