Problem 40 - Precalculus

Question

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

Step-by-Step Solution

Answer

The value of \(\cos(-720°) + \tan(720°)\) is 1.

Step 1: Understanding Angles Beyond 360°

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

Step 2: Evaluating \( \cos(-720°) \)

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

Step 3: Evaluating \( \tan(720°) \)

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

Step 4: Calculating the Expression

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

Key Concepts

Angles Beyond 360°

When dealing with angles, it's important to understand what happens when they exceed 360°. A full rotation in a circle equals 360°. So, any angle greater than 360° can be seen as completing one or more full circles plus a leftover angle.

• For example, the angle 720° means two full rotations around a circle since 720° is double 360°.
• Similarly, a negative angle like -720° implies two complete rotations, but in a clockwise direction.

These angles can be simplified by subtracting or adding multiples of 360° to find an equivalent angle within a single rotation of the circle. For instance:

• For 720°, subtract 360° twice to get 0°, which means it sits at the same position as 0° on the circle.
• The same logic applies to -720°, where adding 360° twice will also yield 0°.

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°, \( \cos(0°) = 1 \), 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°, implying that \( \cos(\theta) = \cos(\theta + 360°k) \) for any integer \(k\).

This means angles like -720° or 720° simplify to the same cosine value as their base angle within the first full rotation, such as 0° in this case.Whether evaluating \( \cos(-720°) \) or \( \cos(720°) \), they both equate to \( 1 \), because their equivalent angle of 0° 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° angle, \(\tan(0°) = 0 \) 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°.

This periodicity allows any angle like 720° to simplify to an equivalent smaller angle by subtracting multiples of 180° until it fits within one cycle of the tangent's period.For example, \( \tan(720°) \) reduces to \(\tan(0°) = 0\), because after subtracting two full circles (720°), the remaining angle is equivalent to 0°, resulting in a tangent value of 0.