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Magazine Chapter 10: Problem 10
Find the exact value or state that it is undefined. $$ \sec \left(-\frac{5 \pi}{3}\right) $$
Short Answer
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Step by step solution
01
Recall the Trigonometric Identity
We know that the secant function is defined as the reciprocal of the cosine function: \( \sec(\theta) = \frac{1}{\cos(\theta)} \). Thus, we need to find \( \cos \left(-\frac{5\pi}{3}\right) \) first.
02
Simplify the Angle
The angle \( -\frac{5\pi}{3} \) is beyond the range of \( 0 \) to \( 2\pi \). To find a coterminal positive angle, add \( 2\pi \):\[-\frac{5\pi}{3} + 2\pi = \frac{6\pi}{3} - \frac{5\pi}{3} = \frac{\pi}{3}.\]
03
Find the Cosine of the Equivalent Angle
Now, we have reduced the angle to \( \frac{\pi}{3} \). We know from the unit circle that \( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \).
04
Calculate the Secant
Use the identity \( \sec(\theta) = \frac{1}{\cos(\theta)} \):\[\sec\left(-\frac{5\pi}{3}\right) = \sec\left(\frac{\pi}{3}\right) = \frac{1}{\cos\left(\frac{\pi}{3}\right)} = \frac{1}{\frac{1}{2}} = 2.\]
05
Conclusion
The value of \( \sec\left(-\frac{5\pi}{3}\right) \) is 2, as calculated above.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Secant Function
The secant function is a trigonometric function that is closely related to the cosine function. To understand secant, it's vital to grasp the basic notion that it is the reciprocal of the cosine.
- The secant of an angle, denoted as \( \sec(\theta) \), is defined by the formula \( \sec(\theta) = \frac{1}{\cos(\theta)} \).
- For every angle where the cosine is non-zero, the secant is simply the inverse of the cosine value.
- Whenever the cosine of an angle is zero, the secant becomes undefined, because division by zero is mathematically indeterminate.
Cosine Function
The cosine function is a fundamental trigonometric function. It's a ratio of the adjacent side to the hypotenuse in a right triangle and is used to describe the shape of a wave-like graph around the unit circle.
- In the unit circle, any point \( (x, y) \) on the circle with radius 1 is described by \( x = \cos(\theta) \), representing the horizontal component of the rotation angle \( \theta \).
- The cosine function's values range from -1 to 1, indicating the maximum and minimum horizontal distances on the unit circle.
- The cosine function is periodic with a period of \( 2\pi \). This means it repeats its values every full circle or 360掳.
Unit Circle
The unit circle is a crucial tool in understanding trigonometry and defining the relationships between different trigonometric functions.
- The unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane.
- It helps visualize how trigonometric functions like sine and cosine are derived from angles, with the coordinates of points on the circle representing the values of these functions.
- Angles on the unit circle are typically measured in radians, with \( 0 \) radians starting from the positive x-axis and moving counter-clockwise.
- Key angles often used are \( \frac{\pi}{6} \), \( \frac{\pi}{4} \), \( \frac{\pi}{3} \), \( \frac{\pi}{2} \), and \( \pi \), with their cosine and sine values commonly memorized to assist in solving trigonometric problems quickly.
