Problem 35 - Algebra 2

Question

Use the graph of the appropriate reciprocal trigonometric function to find each value. Round to four decimal places. $$ \sec 110^{\circ} $$

Step-by-Step Solution

Answer

The value of \(\sec 110^{\circ}\) to four decimal points is approximately computed by this process.

Step 1: Convert to Radians

First, the degree measure has to be converted to radians, since the trigonometric functions in calculators usually operate in radians. Use the formula \(110\degree = 110\degree \times \dfrac{\pi}{180} \) radians.

Step 2: Calculate Cosine

Then, calculate the cosine of that radian measure. To do this, use the 'cos' function on the calculator.

Step 3: Find Reciprocal to get Secant

Finally, to find the secant of the angle, compute the reciprocal of the cosine that was calculated. In mathematical expressions, it can be represented as sec \(\theta = \frac{1}{\cos \theta}\).

Step 4: Round off

Since it is specified to round the result to four decimal places, use the round function to achieve this.

Key Concepts

Secant Function

The secant function is one of the six trigonometric functions and is considered a reciprocal trigonometric function. It is defined as the reciprocal of the cosine function. In mathematical terms, it is expressed as \(\sec \theta = \frac{1}{\cos \theta}\).
This means if we know the value of \(\cos \theta\), we can easily find \(\sec \theta\) by taking its reciprocal or inverse.
The secant function can sometimes be tricky because it is not as commonly used as sine and cosine. However, it is very important in various applications such as engineering and physics.

• The range of the secant function is from \( -\infty \) to \(-1\) and \( 1\) to \(+\infty\), because cosine values range from \(-1\to1\).
• It has vertical asymptotes where the cosine function is zero, as dividing by zero is undefined.
• Understanding secant requires a good grasp of cosine due to their reciprocal relationship.

Radian Conversion

Trigonometric calculations often require angle measures to be in radians rather than degrees. This is because most trigonometric functions on calculators operate in radian mode by default.
To convert an angle from degrees to radians, we use the formula: \(rac{\pi}{180}\).
For example, to convert \(110^\circ\) to radians, we calculate \(110^\circ \times \frac{\pi}{180}\).
Simply multiply the degree measure by \(\frac{\pi}{180}\).

• This formula comes from the fact that \(180\degree\) is equivalent to \(\pi\) radians.
• It's important to make sure your calculator is set to the right mode (radian or degree) to avoid errors in calculations.

Cosine Calculation

Calculating the cosine of an angle is a crucial step in finding the secant. Once the angle is converted to radians, use a calculator to find the cosine value of that radian measure.
Make sure your calculator is set to radian mode.
You can enter the radian measure and use the 'cos' function to get the result.
Suppose you converted \(110^\circ\) to radians and obtained the radian value, simply compute \(\cos\) of that value.

• If \(\theta\) is the angle in radians, \(\cos(\theta)\) gives the x-coordinate of the corresponding point on the unit circle.
• The cosine function is periodic and repeats its values every \(2\pi\) radians.

Round to Four Decimal Places

Rounding numbers can make them easier to use and read. For the purpose of this exercise, the final secant value needs to be rounded to four decimal places.
This means you should look at the fifth decimal place to determine whether to round up or remain the same.
If the fifth decimal digit is 5 or more, round up the fourth decimal by increasing it by one. If it is less than 5, keep the fourth decimal the same.
For example, if your calculator shows \(\sec \theta = 1.234567\), you round it to \(1.2346\).

• Rounding is important for precision, especially in technical and scientific calculations.
• It helps present numbers in a more understandable and manageable format.