Tangent Calculator

Definition and Formula of Tangent

The tangent function, denoted as \(\tan(\theta)\), is a fundamental trigonometric function used to describe the relationship between an angle and the sides of a right triangle. The angle \(\theta\) is usually measured in radians.

In a right triangle, the tangent of an angle \(\theta\) is defined as the ratio of the opposite side to the adjacent side: \( \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{a}{b} \) This represents the vertical to horizontal component ratio of the angle.

On the unit circle, the tangent function is defined as the ratio of the y-coordinate to the x-coordinate of the point corresponding to the angle \(\theta\): \( \tan(\theta) = \frac{y}{x} \) Here, \(x\) and \(y\) are the coordinates of the point on the unit circle.

Examples

Example 1: Calculating Tangent Using a Right Triangle

A right triangle has one acute angle \(\theta = 45^\circ\), with the opposite side measuring 4 units and the adjacent side measuring 4 units. Calculate the tangent of this angle.

Example 2: Tangent in Practical Applications

You observe a tower that is 30 meters tall from a horizontal distance of 40 meters. Calculate the angle \(\theta\) between your line of sight and the ground.

Properties and Graph of Tangent

Key Characteristics:

• Periodicity: The tangent function repeats every \(\pi\) radians (\(180^\circ\)), \(\tan(\theta + n\pi) = \tan(\theta)\), where \(n\) is an integer.
• Monotonicity: The tangent function is monotonically increasing within each period.
• Odd Function: Tangent satisfies \(\tan(-\theta) = -\tan(\theta)\), meaning it is symmetric about the origin.
• Amplitude: The tangent function has no amplitude limit, as its range is all real numbers (\(\mathbb{R}\)).
• Asymptotes: Vertical asymptotes occur at \(\theta = \frac{\pi}{2} + n\pi\), where \(n\) is an integer.
• Domain and Range: The domain of the tangent function is all angles \(\theta\) (except \(\frac{\pi}{2} + n\pi\)), and the range is \(\mathbb{R}\) (all real numbers).

Quadrant Properties

The tangent function's sign varies by quadrant:

Additional Tangent Calculations

1. Reciprocal of Tangent (Cotangent)

The reciprocal of tangent is the cotangent function: \( \frac{1}{\tan(\theta)} = \cot(\theta) = \frac{\text{Adjacent}}{\text{Opposite}} \) Undefined when \(\tan(\theta) = 0\).

2. Derivative of Tangent

The derivative of \(\tan(\theta)\) is the square of the secant function: \( \frac{d}{d\theta} \tan(\theta) = \sec^2(\theta) \) This property is useful in calculus for analyzing rate changes.

3. Integral of Tangent

The integral of \(\tan(\theta)\) is given by: \( \int \tan(\theta) \, d\theta = -\ln|\cos(\theta)| + C \) This integral is frequently used in physics and engineering.

4. Inverse Tangent (arctan)

The arctangent function computes the angle corresponding to a tangent value. Its domain is all real numbers, and its range is \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\): \( \theta = \arctan(x) \)

Common Tangent Values