Problem 45 - Precalculus

Question

Evaluate each expression, if possible. $$\tan 720^{\circ}+\sec 720^{\circ}$$

Step-by-Step Solution

Answer

The result is 1.

Step 1: Simplify Angle for Tangent

Tangent and secant functions have a periodicity of 360°.Thus, \( \tan(720^{\circ}) = \tan(720^{\circ} - 2 \times 360^{\circ}) = \tan(0^{\circ}) \).

Step 2: Evaluate Tangent at 0°

The tangent of 0 degrees is 0. So, \( \tan 0^{\circ} = 0 \).

Step 3: Simplify Angle for Secant

Just like tangent, secant is periodic with a period of 360°.Thus, \( \sec(720^{\circ}) = \sec(720^{\circ} - 2 \times 360^{\circ}) = \sec(0^{\circ}) \).

Step 4: Evaluate Secant at 0°

The secant of 0 degrees is 1, because \( \sec(0^{\circ}) = \frac{1}{\cos(0^{\circ})} \). Since \( \cos(0^{\circ}) = 1 \), \( \sec(0^{\circ}) = 1 \).

Step 5: Combine the Results

Now combine the evaluated results. \( \tan 720^{\circ} + \sec 720^{\circ} = 0 + 1 = 1 \).

Key Concepts

Tangent Function

The tangent function is one of the basic six trigonometric functions. It is represented as \( \tan \theta \) and can be defined as the ratio of the sine and cosine functions: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). The tangent function is particularly useful in situations involving right-angle triangles or when solving problems in the Cartesian plane.

• In a right triangle, it represents the ratio of the length of the opposite side to the adjacent side for a given angle \( \theta \).
• On the unit circle, it reflects the length of the line segment from the origin to the tangent point on the circle.

Understanding \( \tan \theta \) at special angles like \( 0^{\circ}, 90^{\circ}, 180^{\circ}, \) etc., is key to solving trigonometric expressions. For example, the tangent of \( 0^{\circ} \) is 0, as there is no opposite side length relative to the horizontal or x-axis.
Being mindful of the properties and graphs of the tangent function can help simplify complex trigonometric problems.

Secant Function

The secant function is another significant trigonometric function. It is the reciprocal of the cosine function, represented as \( \sec \theta = \frac{1}{\cos \theta} \).The secant function has a few unique properties:

• It provides the ratio of the hypotenuse to the adjacent side in a right triangle.
• It is undefined for angles where \( \cos \theta = 0 \), such as \( 90^{\circ} \) and \( 270^{\circ} \), because division by zero is not possible.

Given its definition, when \( \cos \theta \) is 1 (such as at \( 0^{\circ} \)), the secant function equals 1. Therefore, \( \sec 0^{\circ} = 1 \).
This demonstrates how the secant function can simplify parts of an expression involving cosine.

Periodicity in Trigonometry

Periodicity is a core concept in trigonometry that refers to how certain trigonometric functions repeat their values over specific intervals. For the tangent and secant functions, this repetition happens every \( 360^{\circ} \), or \( 2\pi \) radians. In simpler terms, the functions will start their value cycle once again after completing one full circle on the unit circle measurement.Key points about periodicity:

• For any angle \( \theta \), adding multiples of \( 360^{\circ} \) doesn't change the function's value: \( \tan(\theta + 360^{\circ}k) = \tan \theta \) and \( \sec(\theta + 360^{\circ}k) = \sec \theta \), where \( k \) is an integer.
• This property allows simplification of angles in trigonometric expressions by reducing them to smaller equivalent angles within one period (\( 0^{\circ} \) to \( 360^{\circ} \)).

Understanding periodicity helps simplify calculations and solve trigonometric equations by reducing angles that seem complex into more manageable angles.