Problem 21 - Algebra and Trigonometry

Question

Find the exact value of the trigonometric function. $$ \tan 750^{\circ} $$

Step-by-Step Solution

Answer

The exact value of \( \tan 750^{\circ} \) is \( \frac{1}{\sqrt{3}} \) or \( \frac{\sqrt{3}}{3} \).

Step 1: Understand the Problem

We need to find the exact value of \( \tan 750^{\circ} \). The tangent function is periodic with period \(360^{\circ}\), meaning that the tangent of any angle \( \theta \) is the same as \( \theta + 360^{\circ}k \), where \( k \) is an integer.

Step 2: Reduce the Angle

We will reduce the angle \( 750^{\circ} \) within the first cycle of the tangent's period. Calculate the remainder of \( 750^{\circ} \) divided by \( 360^{\circ} \): \[750^{\circ} - 2 \times 360^{\circ} = 750^{\circ} - 720^{\circ} = 30^{\circ}\] Thus, \( \tan 750^{\circ} = \tan 30^{\circ} \).

Step 3: Evaluate the Tangent Function

Now, evaluate \( \tan 30^{\circ} \). The exact value is known from trigonometric tables or geometric principles: \[\tan 30^{\circ} = \frac{1}{\sqrt{3}}\] This value can also be expressed as \( \frac{\sqrt{3}}{3} \) after rationalizing the denominator.

Key Concepts

Tangent Function

The tangent function, often denoted as \( \tan \), is one of the primary trigonometric functions. It relates the angle of a right triangle to the ratio of the length of the opposite side to the adjacent side.

• The equation for the tangent of an angle \( \theta \) is \( \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} \).
• Tangent is undefined at angles where the cosine is zero because dividing by zero is impossible.
• These angles occur at \( 90^{\circ} \) and \( 270^{\circ} \) within the first cycle.

The tangent function is periodic and repeats its values every \( 180^{\circ} \). This property becomes quite handy when dealing with angles outside the standard range, as seen in the example of \( \tan 750^{\circ} \). It simplifies finding equivalent tangent values for large or negative angles easily, reducing them to their smallest positive equivalent measure.

Angle Reduction

Angle reduction is a crucial technique when solving trigonometric problems with large degree measures. The concept involves reducing a larger angle to an equivalent angle within a basic cycle of the trigonometric function.
This is done by utilizing the periodic properties of these functions. By subtracting an integer multiple of the function's cycle from the original angle, we can find an equivalent angle that lies within one period.

• For instance, in the example of \( 750^{\circ} \), we subtract \( 2 \times 360^{\circ} \), reducing it to \( 30^{\circ} \).
• Since \( 30^{\circ} \) lies within the standard trigonomic interval \( [0^{\circ}, 360^{\circ}) \), it is easier to compute or reference in standard trig tables.

By reducing angles, calculations become much simpler, allowing us to utilize known trigonometric values or identities efficiently and accurately.

Periodicity of Trigonometric Functions

Trigonometric functions demonstrate periodic behavior, which means they exhibit repeated patterns over specific intervals.
For the tangent function, this interval (or period) is \( 180^{\circ} \). This characteristic allows for significant simplification, where any angle measure \( \theta \) is congruent to \( \theta + 180^{\circ}k \) for any integer \( k \).

• Such periodic attributes make it easy to calculate or recognize values beyond the initial \( 360^{\circ} \) range.
• In practice, as shown in the example, reducing \( 750^{\circ} \) to \( 30^{\circ} \) is a straightforward task using the periodic property.
• The converging cycles of tangent mean angles such as \( 750^{\circ} \), \( -150^{\circ} \), and \( 30^{\circ} \) all yield the same tangent value.

By understanding periodicity, one can easily tackle complex trigonometric problems and perform transformations that reveal simpler, equivalent forms regardless of an angle's initial complexity.