Question

Use the fact that the trigonometric functions are periodic to find the exact value of each expression. Do not use a calculator. $$ \sec 540^{\circ} $$

Short answer

\( \sec 540^{\circ} = -1 \)

Step-by-step solution

Understand the Periodicity of the Trigonometric Function

The secant function, \(\text{sec}(x)\), is derived from the cosine function and shares the same periodicity. The cosine function, \(\text{cos}(x)\), has a period of \(\boldsymbol{360^{\boldsymbol{\text{o}}}}\), meaning \(\text{cos}(x+360^{\text{o}}) = \text{cos}(x)\). Therefore, \(\text{sec}(x+360^{\text{o}}) = \text{sec}(x)\).

Reduce the Angle Using Periodicity

To find \(\text{sec}(540^{\text{o}})\), reduce the angle within one period: \(\text{sec}(540^{\text{o}}) = \text{sec}(540^{\text{o}} - 360^{\text{o}}) = \text{sec}(180^{\text{o}})\).

Evaluate the Secant Function at the Reduced Angle

Recall that \(\text{sec}(x) = \frac{1}{\text{cos}(x)}\). So, we need to find \(\text{sec}(180^{\text{o}})\). Since \(\text{cos}(180^{\text{o}}) = -1\), it follows that \(\text{sec}(180^{\text{o}}) = \frac{1}{-1} = -1\).

Key concepts

Periodicity

In trigonometry, periodicity refers to the property of trigonometric functions repeating their values in regular intervals. For the cosine function, \(\text{cos}(x)\), this interval is \(\boldsymbol{360^{\boldsymbol{\text{o}}}}\). This means any angle increased by \(\boldsymbol{360^{\boldsymbol{\text{o}}}}\) will have the same cosine value:
\[ \text{cos}(x+360^{\text{o}}) = \text{cos}(x) \]
Since the secant function, \(\text{sec}(x)\), is defined as \[ \text{sec}(x) = \frac{1}{\text{cos}(x)} \], it inherits this periodicity:
\[ \text{sec}(x+360^{\text{o}}) = \text{sec}(x) \]
This periodic property allows us to simplify angles by subtracting multiples of \(\boldsymbol{360^{\boldsymbol{\text{o}}}}\). By doing this, we reduce complex angles to simpler, equivalent ones within a single period.

Secant Function

The secant function, denoted as \(\text{sec}(x)\), is the reciprocal of the cosine function:
\[ \text{sec}(x) = \frac{1}{\text{cos}(x)} \]
Because of this relationship, \(\text{sec}(x)\) is undefined wherever \(\text{cos}(x)\) is zero, such as at \(\boldsymbol{90^{\boldsymbol{\text{o}}}}\) and \(\boldsymbol{270^{\boldsymbol{\text{o}}}}\). The graph of \(\text{sec}(x)\) has vertical asymptotes at these points.
The secant function shares the same periodicity as the cosine function, repeating every \(\boldsymbol{360^{\boldsymbol{\text{o}}}}\). This property is useful for solving trigonometric problems involving larger angles by reducing them to a simpler equivalent angle within one period, often between \(\boldsymbol{0^{\boldsymbol{\text{o}}}}\) and \(\boldsymbol{360^{\boldsymbol{\text{o}}}}\).

Cosine Function

The cosine function, represented as \(\text{cos}(x)\), is one of the six fundamental trigonometric functions. It relates an angle in a right triangle to the ratio of the adjacent side to the hypotenuse. The cosine function has unique properties:

• Periodic with a period of \(\boldsymbol{360^{\boldsymbol{\text{o}}}}\):
  \[ \text{cos}(x+360^{\text{o}}) = \text{cos}(x) \]
• Even function: \(\text{cos}(-x) = \text{cos}(x)\)
• Range: between -1 and 1

The cosine function is essential for defining other trigonometric functions, including the secant function. For example, to find \(\text{sec}(540^{\boldsymbol{\text{o}}}}\), we use the simplification from cosine function properties decreasing it to within \(\boldsymbol{360^{\boldsymbol{\text{o}}}}\). Then we can find its cosine value.

Angle Reduction

Angle reduction is a technique used in trigonometry to simplify the evaluation of trigonometric functions at larger angles. By using the property of periodicity, large angles can be reduced modulo the period of the function. For instance:
To evaluate \(\text{sec}(540^{\text{o}})\), we reduce \(\boldsymbol{540^{\boldsymbol{\text{o}}}}\) using \(\boldsymbol{360^{\boldsymbol{\text{o}}}}\):
\[ 540^{\text{o}} - 360^{\text{o}} = 180^{\text{o}} \]
Thus, \[ \text{sec}(540^{\text{o}}) = \text{sec}(180^{\text{o}}) \]
This reduced angle, \(\boldsymbol{180^{\boldsymbol{\text{o}}}}\), falls within the first period, making it easier to evaluate. This process ensures that we work with smaller, more manageable angles that fall within a single cycle of the function's period.