Problem 34 - Algebra 2

Question

Use the graph of the appropriate reciprocal trigonometric function to find each value. Round to four decimal places. $$ \sec 80^{\circ} $$

Step-by-Step Solution

Answer

The value of \( \sec(80^\circ)\) is approximately 5.7589 when rounded to four decimal places.

Step 1: Convert degrees to radians

The first step is to convert the given measurement from degrees to radians as this is what our calculators use. The formula for converting degrees to radians is \(radian = degree * (\pi/180)\). So for 80 degrees, it would be \(80 * (\pi/180)\), which is approximately 1.3963 radians.

Step 2: Calculate the cosine

Once we have the measurement in radians, we would then find the cosine of the measurement using a scientific calculator. Enter the number 1.3963 into the calculator and press on the cosine function to find the cosine of 80 degrees, which is approximately 0.1736.

Step 3: Find the reciprocal

The secant function is simply the reciprocal of the cosine function. Therefore, to find the secant, we take the reciprocal of the cosine value. Thus, \( \sec(80^\circ)\) equals to the reciprocal of 0.1736. After calculating, we get an answer of approximately 5.7589 when rounded to four decimal places.

Key Concepts

Secant Function

The secant function, often designated as \( \sec \theta \), is one of the six fundamental trigonometric functions. Trigonometric functions are critical in relating angles to side lengths in right triangles.

The secant function specifically is defined as the reciprocal of the cosine function. When written mathematically, this is expressed as:

• \( \sec \theta = \frac{1}{\cos \theta} \)

If you know the cosine value of an angle, you can easily find the secant by taking its reciprocal. It is important to note that the secant function, similar to cosine, depends on the angle's measurement, which can be in degrees or radians.

Utilizing \( \sec \theta \) can be beneficial in many practical applications like physics, engineering, and even computer graphics when the relationship between angles and distances needs to be determined.

Conversion of Degrees to Radians

Converting from degrees to radians is crucial when dealing with trigonometric functions, especially for scientific calculations. Calculators and many mathematical frameworks use radians instead of degrees as the standard unit of angle measurement.

The formula to convert degrees to radians is:

• \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \)

When you convert an angle from degrees to radians, you multiply the degree value by \( \frac{\pi}{180} \).

For example, converting \( 80^\circ \) to radians involves:

• \( 80 \times \frac{\pi}{180} = 1.3963 \) radians (rounded to four decimal places)

Understanding how to switch between these two units allows you to work more comfortably within different scientific and mathematical contexts. It also helps ensure your calculations align with standard practices in fields that use trigonometry heavily.

Scientific Calculator Use

A scientific calculator is a valuable tool for solving complex mathematical problems, including trigonometric functions. With the push of a button, it can perform operations like sine, cosine, and tangent, along with their reciprocal functions such as secant.

When finding the secant of an angle, the calculator helps you quickly determine its cosine value. For example, if you're calculating \( \cos(80^\circ) \) after converting it to radians, you would:

• Enter the value \( 1.3963 \) into the calculator.
• Press the cosine function button \( \cos \).
• The calculator will display approximately 0.1736, representing the cosine of 80 degrees.

To find the secant, take the reciprocal of the calculator's result, giving you \( \sec(80^\circ) \approx 5.7589 \).

Understanding how to effectively use a scientific calculator simplifies and speeds up problem-solving, making it a crucial skill for students tackling trigonometric problems.