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**Hint:**

Here we will find the value of \[x\] by using trigonometric function identity. First, we will convert the secant function into a cosine function by using the reciprocal trigonometric identity. Then we will find the value of \[\cos x\]. Finally, we will take cosine inverse to get the required answer.

**Complete step by step solution:**

We have to find the value of \[x\] for \[\sec x = - 1.7172\].

We know that secant is also defined as reciprocal of cosine function i.e. \[\sec x = \dfrac{1}{{\cos x}}\].

Therefore, using this identity, we can write

\[\sec x = \dfrac{1}{{\cos x}} = - 1.7172\]

On cross multiplying the terms, we get

\[ \Rightarrow \cos x = \dfrac{1}{{ - 1.7172}}\]

Dividing the terms, we get

\[ \Rightarrow \cos x = - 0.5823\]

Now, taking the inverse cosine function on both the sides, we get

\[ \Rightarrow x = {\cos ^{ - 1}}\left( { - 0.5823} \right)\]

Using the calculator we get,

\[x = {125.61^ \circ }\]

**So, we get the value of \[x\] as \[{125.61^ \circ }\].**

**Additional information:**

The Reciprocal Identity of trigonometric functions states that there are three functions which can be defined as the reciprocal of the other three functions. For example, secant can be defined as reciprocal of cosine, cosecant can be defined as reciprocal of sine and cotangent can be defined as reciprocal of the tangent.

**Note:**

Trigonometry is that branch of mathematics that deals with specific functions of angles and also their application in calculations and simplification. The commonly used six types of trigonometry functions are defined as sine, cosine, tangent, cotangent, secant and cosecant. Identities are those equations which are true for every variable. The trigonometric functions are those real functions that relate the angle to the ratio of two sides of a right-angled triangle.

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