Question

Evaluate the function without using a calculator. \(\sec \frac{11 \pi}{6}\)

Short answer

The value of \( \sec \frac{11 \pi}{6} \) is \( 2\sqrt{3}/3 \)

Step-by-step solution

Understand the problem

The sec function is the reciprocal of the cos function, that means: \( \sec(x) = 1/cos(x) \). We are looking for the value of \( \sec \frac{11 \pi}{6} \) which is equal to \( 1/cos \frac{11 \pi}{6} \).

Find the value for cosine of the angle

First, let's find the value for \( cos(\frac{11 \pi}{6}) \). Looking at the unit circle, \( cos(\frac{11 \pi}{6}) = cos(\frac{\pi}{6}) \) because they are coterminal angles (angles with the same terminal side). From elementary trigonometry, we know that \( cos(\frac{\pi}{6}) = \sqrt{3}/2 \).

Find the value for the sec function

Now that we know the value of the cosine function, we can find the secant. Remember, the secant is the reciprocal of cosine, so \( \sec \frac{11 \pi}{6} = 1/cos \frac{11 \pi}{6} = 1/(\sqrt{3}/2) = 2/\sqrt{3} \). This is an acceptable answer, but we might want to rationalize the denominator to be \( 2\sqrt{3}/3 \).

Key concepts

Secant Function

The secant function, often abbreviated as "sec," is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the cosine function.
This means that for any angle \( x \), the secant function is expressed as:

• \( \sec(x) = \frac{1}{\cos(x)} \)

In trigonometry, secant is particularly useful in scenarios where the cosine of an angle is either very small or zero. It magnifies these values, making it easier to work with small angles or angles near zero.
To evaluate the secant function without a calculator, it's essential to first determine the cosine value for the angle involved and then compute its reciprocal. This process allows for determining the sec value manually, which is often required in exam settings.

Unit Circle

The unit circle is a critical concept in trigonometry and geometry. It is a circle with a radius of one unit centered at the origin of the coordinate system. This circle is particularly useful for defining trigonometric functions for all angles.
Using the unit circle, any angle can be represented as a point on this circle, where the x-coordinate of the point is the cosine of the angle, and the y-coordinate is the sine of the angle.

• The unit circle aids in understanding angles in both degrees and radians, providing insight into their periodic nature.
• It helps visualize how the trigonometric functions behave as angles increase or decrease.

By using the unit circle, one can easily see how angles such as \( \frac{11 \pi}{6} \) relate to more familiar angles like \( \frac {\pi}{6} \). This facilitates understanding coterminal angles and helps in evaluating trigonometric functions without a calculator.

Reciprocal Function

The concept of reciprocal functions plays a vital role in mathematics, particularly in trigonometry. A reciprocal function involves taking the reciprocal of a given function, effectively flipping the fraction of an equation.
For trigonometric functions, the basic reciprocal functions are:

• Secant (sec), which is the reciprocal of cosine (cos): \( \sec(x) = \frac{1}{\cos(x)} \)
• Cosecant (csc), which is the reciprocal of sine (sin): \( \csc(x) = \frac{1}{\sin(x)} \)
• Cotangent (cot), which is the reciprocal of tangent (tan): \( \cot(x) = \frac{1}{\tan(x)} \)

These reciprocal relationships are crucial for solving various trigonometric problems, especially when certain trigonometric values are required but not directly available.
Knowing how to switch between a function and its reciprocal can dramatically simplify trigonometric calculations and is fundamental in manipulating trigonometric identities.

Coterminal Angles

Coterminal angles are angles that share the same terminal side when drawn in standard position on a circle. This means they end at the same position on the unit circle but have different measures.
Coterminal angles can be found by adding or subtracting full rotations from an original angle. Since one full rotation is \( 2\pi \) radians (or 360 degrees), two angles are coterminal if they differ by \( 2\pi \) times \( n \), where \( n \) is an integer.

• For example, \( \pi \) radians and \( 3\pi \) radians are coterminal because \( 3\pi - \pi = 2\pi \).
• This relationship helps simplify trigonometric calculations by allowing us to use the properties of familiar angles to solve problems involving less familiar angles.

Understanding coterminal angles is crucial in trigonometry as it allows for the evaluation of trigonometric functions like secant or cosine at angles like \( \frac{11\pi}{6} \), by recognizing that it is equivalent to simpler angles such as \( \frac{\pi}{6} \). This makes evaluations without a calculator significantly easier.