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Chapter 4: Problem 40

Evaluate each expression, if possible. $$\cos \left(-720^{\circ}\right)+\tan 720^{\circ}$$

Short Answer

Expert verified
The value of \(\cos(-720掳) + \tan(720掳)\) is 1.

Step by step solution

01

Understanding Angles Beyond 360掳

Angles larger than 360掳 correspond to one or more full circles plus an additional angle. So, a negative angle of -720掳 is equivalent to rotating clockwise by two full circles (since each circle is 360掳).Similarly, a positive angle of 720掳 represents two full counter-clockwise circles. Thus,\(-720掳 = 0掳\) and \(720掳 = 0掳\).
02

Evaluating \( \cos(-720掳) \)

Since \(-720掳\) simplifies to \(0掳\), we need to evaluate \(\cos(0掳)\). The cosine of 0掳 is 1, so \(\cos(-720掳) = 1.\)
03

Evaluating \( \tan(720掳) \)

Similarly, simplify \(720掳\) to \(0掳\). The tangent of 0掳 is defined as 0, therefore \(\tan(720掳) = 0.\)
04

Calculating the Expression

Add the results from steps 2 and 3: \(\cos(-720掳) + \tan(720掳) = 1 + 0 = 1.\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Angles Beyond 360掳
When dealing with angles, it's important to understand what happens when they exceed 360掳. A full rotation in a circle equals 360掳. So, any angle greater than 360掳 can be seen as completing one or more full circles plus a leftover angle.
  • For example, the angle 720掳 means two full rotations around a circle since 720掳 is double 360掳.
  • Similarly, a negative angle like -720掳 implies two complete rotations, but in a clockwise direction.
These angles can be simplified by subtracting or adding multiples of 360掳 to find an equivalent angle within a single rotation of the circle. For instance:
  • For 720掳, subtract 360掳 twice to get 0掳, which means it sits at the same position as 0掳 on the circle.
  • The same logic applies to -720掳, where adding 360掳 twice will also yield 0掳.
This simplification helps in evaluating trigonometric functions like cosine and tangent.
Cosine Function
The cosine function is one of the primary trigonometric functions. It relates an angle to the x-coordinate of a point on the unit circle. Specifically, cosine measures how far left or right the point is along the horizontal axis.
  • The cosine of 0掳, \( \cos(0掳) = 1 \), represents the maximum value to the right, meaning the point is fully extended along the positive x-axis.
  • The function is periodic, with a period of 360掳, implying that \( \cos(\theta) = \cos(\theta + 360掳k) \) for any integer \(k\).
This means angles like -720掳 or 720掳 simplify to the same cosine value as their base angle within the first full rotation, such as 0掳 in this case.Whether evaluating \( \cos(-720掳) \) or \( \cos(720掳) \), they both equate to \( 1 \), because their equivalent angle of 0掳 on the unit circle gives a cosine value of 1.
Tangent Function
The tangent function is another key trigonometric function. It relates to the y-coordinate divided by the x-coordinate on the unit circle, essentially measuring the slope of the angle's terminal side.
  • For a 0掳 angle, \(\tan(0掳) = 0 \) due to it having no slope, corresponding to a flat line along the positive x-axis.
  • Similar to the cosine, the tangent function is periodic. However, its period is shorter, repeating every 180掳.
This periodicity allows any angle like 720掳 to simplify to an equivalent smaller angle by subtracting multiples of 180掳 until it fits within one cycle of the tangent's period.For example, \( \tan(720掳) \) reduces to \(\tan(0掳) = 0\), because after subtracting two full circles (720掳), the remaining angle is equivalent to 0掳, resulting in a tangent value of 0.

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Most popular questions from this chapter

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