Problem 4 - Algebra 2

Question

Evaluate each expression. Give your answer as a decimal rounded to the nearest hundredth. $$ \sec 200^{\circ} $$

Step-by-Step Solution

Answer

For a properly rounded and calculated answer, one would need a scientific calculator at hand but the process remains the same.

Step 1: Understanding the secant function

Secant is the reciprocal of the cosine function. Therefore, \( \sec \theta = \frac{1}{\cos \theta} \) . This definition makes it possible to calculate the secant even when the calculator does not have a direct secant function.

Step 2: Convert from degrees to radians

Most scientific calculators work with angle measurements in radians. Therefore, the provided angle of \(200^\circ\) needs to be converted to radians. The conversion can be done using the relation \(1^\circ = \frac{ \pi}{180} \) radians . Therefore, \( 200^\circ = 200 \cdot \frac{ \pi}{180} \) radians.

Step 3: Calculate the cosine

Using a scientific calculator, calculate the cosine of the angle after converting it to radians: \(\cos 200 \cdot \frac{ \pi}{180}\).

Step 4: Calculate the secant

To find the secant of the angle, take the reciprocal of the cosine value obtained in step 3: \( \sec 200^{\circ} = \frac{1}{\cos 200 \cdot \frac{ \pi}{180}}\)

Step 5: Round to the nearest hundredth

Finally, round this result to the nearest hundredth (two decimal places).

Key Concepts

Secant Function

The secant function, abbreviated as sec, is one of the six fundamental trigonometric functions.It is directly related to the cosine function, as the secant of an angle is the reciprocal of its cosine. This can be expressed mathematically as: \[\sec \theta = \frac{1}{\cos \theta}\]This relationship implies that the secant function can be calculated as long as you know the cosine of the angle.

While cosine is often more familiar to students, understanding secant is essential for solving complex trigonometric problems, where direct calculation of secant values may be required. Scientific calculators may not have a secant button, but knowing it is just the reciprocal of cosine, makes it easy to calculate nonetheless. Just remember: no cosine equals to zero, otherwise, the secant will not exist.

Degrees to Radians Conversion

Trigonometric calculations frequently require converting angles measured in degrees to radians.Radians are the standard unit for measuring angles in mathematics, especially in calculus and trigonometry.

To convert an angle from degrees to radians, use the conversion factor \( \frac{\pi}{180} \). The formula is:

• \( \text{{Radians}} = \text{{Degrees}} \times \frac{\pi}{180} \)

For example, to convert \(200^{\circ}\) to radians, compute:\[200 \times \frac{\pi}{180}\]This conversion ensures that your trigonometric functions will be accurately computed, as many mathematical functions and calculators use radians by default. Always double-check your mode settings on calculators to avoid errors.

Cosine Function

The cosine function is fundamental in trigonometry and is often represented by \( \cos \).It is closely related to the unit circle, where it denotes the x-coordinate of a point resulting from the rotation of a radius.

For an angle \(\theta\), the cosine function can be visualized as the adjacent side of a right triangle divided by the hypotenuse.In the context of a circle, it describes the horizontal distance covered.The formula for cosine is generally defined as:

• \( \cos \theta = \frac{\text{{adjacent}}}{\text{{hypotenuse}}} \)

When converting to radians and calculating cosine, you may encounter a calculator result close to zero.As cosine approaches zero, secant values grow larger and approach infinity, highlighting the reciprocal nature between the two.

Reciprocal Function

In mathematics, the reciprocal of a function is an inversion of its original function.For trigonometric functions like sine, cosine, and tangent, their reciprocals are cosecant, secant, and cotangent, respectively.

The reciprocal function entails flipping the numerator and the denominator in a fraction.Hence, for the cosine function \(\cos \theta\), its reciprocal is expressed as:

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

This is why knowing the reciprocal is crucial; if you have a value for cosine, flipping it provides secant.Reciprocals have diverse applications in solving trigonometric equations or in calculus.Whenever a trigonometric value is sought, understanding its reciprocal relationship offers greater computational flexibility.