Problem 21 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

\(\tan 750^{\circ} = \frac{1}{\sqrt{3}}\).

Step 1: Determine Coterminal Angle

The first step is to find an equivalent angle between \(0^{\circ}\) and \(360^{\circ}\) by subtracting \(360^{\circ}\) multiples from \(750^{\circ}\). Start with \(750^{\circ} - 360^{\circ} = 390^{\circ}\). Then subtract \(360^{\circ}\) again: \(390^{\circ} - 360^{\circ} = 30^{\circ}\). Hence, the coterminal angle is \(30^{\circ}\).

Step 2: Recall the Tangent Value at Reference Angle

Now, apply the tangent function to the coterminal angle found. Recall that \(\tan(30^{\circ}) = \frac{1}{\sqrt{3}}\). This value comes from the common trigonometric ratio for a \(30^{\circ}\) angle in a 30-60-90 triangle.

Key Concepts

Coterminal Angles

Coterminal angles are angles that share the same terminal side but are measured differently because they are obtained by adding or subtracting full rotations (multiples of 360 degrees) from each other. This is an essential concept in trigonometry because it helps to simplify the process of working with angles that are either too large or too small. In order to find coterminal angles, you can add or subtract 360 degrees from the given angle as many times as needed to land an equivalent angle within the more manageable range of 0 to 360 degrees.

• To find a coterminal angle, subtract 360 degrees until the angle is between 0 and 360 degrees.
• Alternatively, you can add 360 degrees to a negative angle to bring it into a positive range.

In the exercise, we took a large angle, 750 degrees, and converted it to a smaller, coterminal angle, 30 degrees, by subtracting 360 degrees twice.

Tangent Function

The tangent function is a fundamental trigonometric function that is essential for solving problems in geometry and trigonometry. It is defined as the ratio of the opposite side to the adjacent side in a right triangle when looking from a given angle. This function is periodic, which means its values repeat at regular intervals as the angle increases.

• The tangent of an angle can be expressed as \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).
• Tangent is positive in the first and third quadrants, where both sine and cosine have the same sign.

In the process of solving for \( \tan(750^{\circ}) \), once the coterminal angle \(30^{\circ}\) is identified, it becomes straightforward to use the known tangent value: \( \tan(30^{\circ}) = \frac{1}{\sqrt{3}} \).

30-60-90 Triangles

A 30-60-90 triangle is a special type of right triangle where the angles measure 30, 60, and 90 degrees. This triangle has a consistent ratio across its sides, allowing for easy calculations of trigonometric functions for these specific angles. This makes it an invaluable shape for understanding trigonometry basics.

• The side opposite the 30-degree angle is the shortest and is usually noted as \(x\).
• The side opposite the 60-degree angle is \(x\sqrt{3}\).
• The side opposite the 90-degree angle is \(2x\), which is the hypotenuse.

These side ratios are the basis for memorizing common trigonometric values for 30° and 60°. For instance, the tangent of a 30-degree angle depends on these side ratios and is calculated as \( \tan(30^{\circ}) = \frac{1}{\sqrt{3}} \), which arises from the opposite side length (\(x\)) divided by the adjacent side length (\(x\sqrt{3}\)). This makes solving trigonometric functions at these angles quick and efficient.