Problem 36 - Algebra 2

Question

Find each exact value. Use a sum or difference identity. $$ \tan 390^{\circ} $$

Step-by-Step Solution

Answer

The exact value of \( \tan(390^{\circ}) \) is \( \frac{1}{\sqrt{3}} \)

Step 1: Convert the angle in range

Given angle is 390 degrees. Since the period of tan function is 360 degrees, by subtracting 360 degrees from the given angle, we obtain an angle within the first revolution: \( 390^{\circ} - 360^\circ = 30^\circ \)

Step 2: Find the exact value

The exact value of tan at 30 degrees can be easily identified from the unit circle or from standard trigonometric values. So, \( \tan(30^{\circ}) = \frac{1}{\sqrt{3}} \)

Key Concepts

Sum and Difference Identities

Sum and difference identities are fundamental in trigonometry. They help express trigonometric functions of sums or differences of angles, using simpler functions of individual angles. These identities are essential for simplifying expressions and solving equations. For the tangent function, the sum identity is:

• \( \tan(a + b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)} \)

Similarly, for the difference of angles, we have:

• \( \tan(a - b) = \frac{\tan(a) - \tan(b)}{1 + \tan(a)\tan(b)} \)

The use of these identities allows us to break down complex angle expressions into manageable parts, making it easier to find exact values. In this exercise, it's beneficial to understand these identities though we relied more directly on the simplified form of the tangent value.

Unit Circle

The unit circle is a circle with radius 1 centered at the origin of a coordinate plane. It’s a fundamental tool in trigonometry because it helps visualize angles and their corresponding trigonometric values. On the unit circle:

• The point \((1, 0)\) represents an angle of \(0^{\circ}\) or \(360^{\circ}\).
• The point \((\frac{\sqrt{3}}{2}, \frac{1}{2})\) corresponds to a \(30^{\circ}\) angle.

Using the unit circle, you can easily see that the tangent of an angle is the ratio of the y-coordinate to the x-coordinate of the point on the circle at that angle. This visualization helps in understanding why \( \tan(30^{\circ}) = \frac{1}{\sqrt{3}} \).

Angle Conversion

Angle conversion is about changing angles to a different form while maintaining their value. It's often necessary to convert angles to fit within the standard range of \([0^{\circ}, 360^{\circ})\) or \([0, 2\pi)\) radians, especially when using periodic functions like tangent. When given an angle like \(390^{\circ}\):

• Subtract \(360^{\circ}\) to find a coterminal angle within the desired range.
• Here, \( \tan(390^{\circ}) \) gets simplified to \( \tan(30^{\circ}) \) after conversion.

This process simplifies the problem and makes it easier to use known trigonometric values.

Tangent Function

The tangent function, denoted \( \tan(\theta) \), relates the angles of a right triangle to the ratio of the side lengths. Specifically, \( \tan(\theta) \) is the ratio of the opposite side to the adjacent side.Characteristics include:

• Periodic with a period of \(180^{\circ}\) or \(\pi\) radians, meaning it repeats its values every \(180^{\circ}\).
• Can take any real number value since there are angles where tangent is undefined (asymptotes).
• At \(30^{\circ}\), \( \tan(30^{\circ}) = \frac{1}{\sqrt{3}} \), a value often used in trigonometry.

This makes the tangent function versatile for solving many trigonometric equations, simplifying complex expressions, and determining exact values like in this exercise.