Question

Use a calculator to approximate the value. Round your answer to two decimal places. $$ \operatorname{arcsec} 1.269 $$

Short answer

The approximate value of \( \operatorname{arcsec} 1.269 \) is 38.42°

Step-by-step solution

Find Reciprocal

First calculate the reciprocal of 1.269. This is done by dividing 1 by 1.269. Using a calculator gives approximately 0.787.

Calculate Arccos

Then, use your calculator to find the cosine inverse or arccos of this value, 0.787. Make sure your calculator is set to the right mode (typically degrees) for this calculation. This will give you the angle whose secant is 1.269.

Round Value

Finally, round the result to two decimal places, as per the instructions in the problem. This gives your final solution.

Key concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, also known as arc functions, are the reverse processes of the trigonometric functions. They are used to determine the angles when the sides of a right-angled triangle are known. For example, the arcsecant function, denoted as \(\operatorname{arcsec}\), is the inverse of the secant function. The output of \(\operatorname{arcsec}(x)\) is an angle whose secant is \(x\). It's important to remember that these functions only provide an angle within a certain range, which for arcsecant is typically \(0 \leq \theta \leq \pi\), excluding \(\pi/2\) when working in radians.

Other common inverse trigonometric functions include arcsine (\(\operatorname{arcsin}\)), arccosine (\(\operatorname{arccos}\)), and arctangent (\(\operatorname{arctan}\)). When using inverse trigonometric functions, it's essential to understand their domain and range to obtain the correct angle.

Reciprocal Calculation

Reciprocal calculation is simply taking the inverse of a given number. In mathematics, the reciprocal of a number \(x\) is \(1/x\). It can also be thought of as dividing 1 by that number. When dealing with trigonometric functions, the concept of reciprocals becomes very useful, especially when converting between functions like secant and cosine, or cosecant and sine. The secant of an angle, for example, is the reciprocal of the cosine of that angle. Therefore, to find the arcsecant of a number, one must first find its reciprocal which will then be applied to the cosine function to determine the corresponding angle.

Arccosine Function

The arccosine function, denoted as \(\operatorname{arccos}\), is the inverse of the cosine function. It's used to find an angle when the cosine of that angle is known. The output of the \(\operatorname{arccos}\) function is the range of angles from \(0\) to \(\pi\) radians or \(0^\circ\) to \(180^\circ\) when working in degrees. When you input the reciprocal of a number into the arccosine function, you're essentially finding the angle whose cosine equals that reciprocal value. In the context of the arcsecant calculation, finding the arccosine of a number's reciprocal provides you with the angle whose secant (which is reciprocal of cosine) equals the original number.

Rounding Decimal Places

Rounding decimal places is the process of shortening a number to a desired number of digits after the decimal point, often for simplicity or to match a specified precision. When rounding to two decimal places, you look at the third decimal digit. If this digit is five or greater, you increase the second decimal digit by one. If it's less than five, you leave the second decimal digit unchanged. In mathematical exercises, like calculating arcsecant values, rounding is a crucial final step for presenting the answer in a clear and concise manner that aligns with the problem's requested precision.