Problem 13 - Algebra and Trigonometry

Question

9–32 Find the exact value of the trigonometric function. $$\tan \left(-60^{\circ}\right)$$

Step-by-Step Solution

Answer

\( \tan(-60^{\circ}) = -\sqrt{3} \).

Step 1: Understand the Tangent Function

The tangent function, \( \tan(\theta) \), is equal to the sine of \( \theta \) divided by the cosine of \( \theta \): \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \). We will apply this understanding to evaluate \( \tan(-60^{\circ}) \).

Step 2: Recognize the Negative Angle Property

For a trigonometric function, \( \tan(-\theta) = -\tan(\theta) \). This property tells us that the tangent of a negative angle is the negative of the tangent of its positive counterpart. This simplifies our calculation to finding \(-\tan(60^{\circ})\).

Step 3: Evaluate \(\tan(60^{\circ})\)

Recall that \(60^{\circ}\) is one of the standard angles with known trigonometric values. For \(\tan(60^{\circ})\), we have \(\tan(60^{\circ}) = \sqrt{3}\).

Step 4: Calculate \(-\tan(60^{\circ})\)

Using the results from Steps 2 and 3, we find that \( \tan(-60^{\circ}) = -\tan(60^{\circ}) = -\sqrt{3} \).

Key Concepts

Negative Angle Property

When working with negative angles in trigonometry, it's important to understand the negative angle property. For the tangent function, this property states that \( \tan(-\theta) = -\tan(\theta) \). This means that to find the tangent of a negative angle, you can simply find the tangent of the positive angle and then negate it.

This property helps simplify calculations. Instead of dealing with the trigonometric function of a negative angle directly, you can convert it to a more familiar positive angle. For example, finding \( \tan(-60^{\circ}) \) becomes much easier when you realize it's just \(-\tan(60^{\circ}) \).

Understanding the negative angle property is useful not just for tangent but also for other trigonometric functions. However, while cosine of negative angles remains the same \( \cos(-\theta) = \cos(\theta) \), sine and tangent both flip sign.

Trigonometric Identities

Trigonometric identities are fundamental relationships between trigonometric functions that allow you to simplify and solve more complex equations.

One of the basic identities involves the tangent function, which is defined as the ratio of sine to cosine: \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \). This relationship is crucial when working with compound angles or simplifying trigonometric expressions.

• Reciprocal identities: These link trigonometric functions with their reciprocals, like \( \tan(\theta) = \frac{1}{\cot(\theta)} \).
• Pythagorean identities: These relate the square of sine, cosine, and tangent, such as \( 1 + \tan^2(\theta) = \sec^2(\theta) \).

Using these identities can help in transforming expressions and solving problems where direct calculation isn't straightforward.

Standard Angles

Standard angles are specific angles that have well-known and frequently used trigonometric values. Understanding these is helpful in solving problems quickly without needing a calculator.

• Angles like \(30^{\circ}, 45^{\circ}, 60^{\circ}\), and \(90^{\circ}\) are considered standard angles.
• The trigonometric functions of these angles often appear in textbook problems.

For example, knowing that \( \tan(60^{\circ}) = \sqrt{3} \) is one such value you should be familiar with. These values are often derived from geometric properties, like the properties of a 30-60-90 triangle.

By memorizing the trigonometric values of standard angles, you'll be able to solve many exercises quickly and efficiently, as seen with \( \tan(-60^{\circ}) \), which uses the value \( \tan(60^{\circ}) \).