Inverse trigonometric functions, also known as arc functions, are the inverses of the sine, cosine, and tangent functions. They are used to determine the angles from given trigonometric ratios. These functions are pivotal in fields like geometry, physics, and engineering, and are a fundamental part of the A-Level mathematics curriculum. Understanding the concept, range, and application of these functions is crucial for students.

**Inverse Sine Function (Arcsine)**

**Definition:**

- The inverse sine function, denoted as $\sin^{-1}x$ or $arcsin(x)$, is defined as the inverse of the sine function.
**Purpose**: It is used to determine the angle whose sine is a known value.**Mathematical Operation**: Given a sine value, the inverse sine function returns the corresponding angle.

**Domain and Range:**

**Domain**: The function is defined for all real numbers $x$within the range $[-1, 1]$. This range corresponds to the possible outputs of the sine function, as sine values always fall between -1 and 1.**Range**: The output of the inverse sine function, which represents angles, is in the range $[- \frac{\pi}{2}, \frac{\pi}{2}]$, or in degrees, between -90° and 90°.

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**Example**

Find the angle $\theta$ if $\sin(\theta) = 0.5$.

**Solution:**

Use the inverse sine function, $\sin^{-1},$ to find $\theta$.

- Calculation: $\theta = \sin^{-1}(0.5)$.
- The angle with a sine of 0.5 is $\frac{\pi}{6}$ or 30 degrees.

**Answer:** $\theta = \frac{\pi}{6}$ or 30 degrees.

**Inverse Cosine Function (Arccosine)**

The inverse cosine function, denoted as ( \cos^{-1}x ) or arccos(x), is the inverse of the standard cosine function.

**Definition:**

- The inverse cosine function, denoted as $\cos^{-1}x$ or $arccos(x)$, is the inverse of the standard cosine function.
**Purpose**: It is used to find the angle whose cosine is a known number, essentially reversing the operation of the cosine function.**Mathematical Operation**: Given a cosine value, the inverse cosine function returns the corresponding angle.

**Domain and Range:**

**Domain**: This function is defined for all real numbers $x$ within the range $[-1, 1]$. This range corresponds to the possible outputs of the cosine function, since cosine values range from -1 to 1.**Range**: The range of the inverse cosine function is $[0, \pi]$ radians, or in degrees, from 0° to 180°. This range is chosen to provide a unique angle for every cosine value.

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**Example**

Determine $\theta$ for $\cos(\theta) = 0.5$.

**Solution:**

Use the inverse cosine function, $\cos^{-1}$, to find $\theta$.

- Calculation: $\theta = \cos^{-1}(0.5)$.
- The angle with a cosine of 0.5 is $\frac{\pi}{3}$ or 60 degrees.

**Answer:** $\theta = \frac{\pi}{3}$ or 60 degrees.

**Inverse Tangent Function (Arctangent)**

**Definition:**

**Inverse of Tangent**: The inverse tangent function is the inverse of the standard tangent function.**Purpose**: It is used to determine the angle whose tangent (the ratio of the opposite side to the adjacent side in a right-angled triangle) is a known value.**Mathematical Operation**: Given a tangent value, the inverse tangent function provides the corresponding angle.

**Domain and Range:**

**Domain**: Unlike the sine and cosine functions, the domain of the inverse tangent function is all real numbers. This is because the tangent function, which relates the ratio of two sides of a right triangle, can take any real value.**Range**: The range of the inverse tangent function is $[- \frac{\pi}{2}, \frac{\pi}{2}]$radians, or in degrees, from -90° to 90°. This range ensures that each tangent value corresponds to one unique angle.

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**Example**

Calculate $\theta$ when $\tan(\theta) = 1$.

**Solution:**Use the inverse tangent function, $\tan^{-1}$, to find $\theta$.- Calculation: $\theta = \tan^{-1}(1)$.
- The angle with a tangent of 1 is $\frac{\pi}{4}$ or 45 degrees.

**Answer:** $\theta = \frac{\pi}{4}$ or 45 degrees.