# sec^{−1} Calculator

x= hypotenuse/adjacent_side

sec^{−1}(x)

## Calculating Inverse Secant of a Value

The inverse secant (sec⁻¹) function, also known as arcsecant, is used to find the angle whose secant is a given value. This function is useful in trigonometry when you need to determine the angle based on the ratio of the hypotenuse to the adjacent side in a right-angled triangle.

The inverse secant of a value \( x \) is defined as:

\( \theta = \sec^{-1}(x) \)

where:

**θ**is the angle corresponding to the secant value.**x**is the secant value, representing the ratio of the hypotenuse to the adjacent side.

Consider the following triangle ABC, right-angled at vertex B.

For the angle \( \theta \) at vertex A:

- Hypotenuse = AC
- Adjacent side = AB

Therefore,

\( \sec^{-1}(x) = \theta \) where \( x = \dfrac{\text{length of side AC}}{\text{length of side AB}} \)

## Examples

The following examples demonstrate how to use the inverse secant function to find the angle when the secant value is known.

### 1. A radio tower is supported by a guy wire that is 50 meters long and is attached to the ground 30 meters from the base of the tower. What is the angle between the guy wire and the ground?

#### Answer

Given:

- Length of the hypotenuse (guy wire) = 50 meters
- Length of the adjacent side (distance from the base of the tower) = 30 meters

First, calculate the secant of the angle:

\( \sec(\theta) = \dfrac{\text{length of guy wire}}{\text{distance from base}} = \dfrac{50}{30} \)

Simplify the expression:

\( \sec(\theta) \approx 1.67 \)

Now, find the angle using the inverse secant function:

\( \theta = \sec^{-1}(1.67) \)

Using a calculator or reference table, the angle is:

∴ θ ≈ 53.1°

### 2. A ramp leads up to a building's entrance. The length of the ramp is 12 meters, and the horizontal distance it covers is 10 meters. What is the angle of elevation of the ramp?

#### Answer

Given:

- Length of the hypotenuse (ramp) = 12 meters
- Length of the adjacent side (horizontal distance) = 10 meters

First, calculate the secant of the angle:

\( \sec(\theta) = \dfrac{\text{length of ramp}}{\text{horizontal distance}} = \dfrac{12}{10} \)

Simplify the expression:

\( \sec(\theta) = 1.2 \)

Now, find the angle using the inverse secant function:

\( \theta = \sec^{-1}(1.2) \)

Using a calculator or reference table, the angle is:

∴ θ ≈ 33.6°

### 3. A crane's cable is extended to lift a load. The cable is 25 meters long and is anchored 20 meters horizontally from the crane's base. What is the angle between the cable and the ground?

#### Answer

Given:

- Length of the hypotenuse (cable) = 25 meters
- Length of the adjacent side (horizontal distance) = 20 meters

First, calculate the secant of the angle:

\( \sec(\theta) = \dfrac{\text{length of cable}}{\text{horizontal distance}} = \dfrac{25}{20} \)

Simplify the expression:

\( \sec(\theta) = 1.25 \)

Now, find the angle using the inverse secant function:

\( \theta = \sec^{-1}(1.25) \)

Using a calculator or reference table, the angle is:

∴ θ ≈ 36.9°

### 4. A waterslide descends at a steep angle into a pool. The slide is 40 meters long, and the distance from the start of the slide to the pool along the ground is 30 meters. What is the angle of the slide with the horizontal?

#### Answer

Given:

- Length of the hypotenuse (slide) = 40 meters
- Length of the adjacent side (horizontal distance) = 30 meters

First, calculate the secant of the angle:

\( \sec(\theta) = \dfrac{\text{length of slide}}{\text{horizontal distance}} = \dfrac{40}{30} \)

Simplify the expression:

\( \sec(\theta) \approx 1.33 \)

Now, find the angle using the inverse secant function:

\( \theta = \sec^{-1}(1.33) \)

Using a calculator or reference table, the angle is:

∴ θ ≈ 41.8°