#### Tan 85 Degrees

**Answer:**

**Introduction to Tangent:**

In trigonometry, the tangent function is one of the fundamental trigonometric functions. It relates the angle of a right triangle to the ratio of the length of the opposite side to the length of the adjacent side. The tangent function is denoted as tan.

In this lesson, we will focus on understanding the tangent function and specifically calculate the tangent of 85 degrees.

**Understanding Tangent:**

Before we dive into calculating the tangent of 85 degrees, let’s first understand what the tangent function represents.

The tangent of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. Mathematically, it is defined as:

Here, Opposite refers to the side of the triangle opposite to the angle, and adjacent refers to the side adjacent to the angle.

**Calculating Tan 85 Degrees:**

To calculate the tangent of 85 degrees, we first need to identify the opposite and adjacent sides in a right triangle. Since we don’t have a specific triangle, we will consider an arbitrary right triangle for our calculation.

Let’s assume we have a right triangle ABC, where angle B is 90 degrees and angle A is 85 degrees. Side BC is the hypotenuse, side AC is the opposite side, and side AB is the adjacent side to angle A.

Now, to calculate the tangent of 85 degrees, we will use the tangent function:

In our example triangle ABC, the opposite side is AC, and the adjacent side is AB. Let’s assume AC = 5 units and AB = 3 units.

Substituting the values into the formula, we have:

Now, we need to evaluate the numerical value of the tangent of 85 degrees. To do this, we can use a calculator or look up the value in a tangent table. Using a calculator, we find that:

tan(85) ≈ 11.4301

Therefore, the approximate value of tan(85) is 11.4301.

**Common Mistakes and Tips:**

When calculating the tangent of an angle, it’s important to remember a few common mistakes and tips:

- Make sure you use the correct formula:
- Double-check your calculation and ensure you substitute the correct values for the opposite and adjacent sides.
- Always verify if your calculator is set to degrees or radians mode. The tangent function expects an angle in degrees, so it’s crucial to use the correct mode.
- Be cautious when dealing with angles close to 90 degrees. The tangent function becomes very large or undefined as the angle approaches 90 degrees.

**Real-World Applications:**

The tangent function is widely used in various fields such as engineering, physics, and navigation. Here are a few real-world applications where the tangent function plays a significant role:

- Architecture: Architects use the tangent function to calculate the angles of inclination for ramps, staircases, and roofs.
- Astronomy: In astronomical calculations, the tangent function helps determine the distances between celestial objects based on the angles observed.
- Engineering: Engineers rely on the tangent function to calculate the angles of slopes, forces applied at an angle, and mechanical leverage systems.
- Surveying: Surveyors utilize the tangent function to measure heights, distances, and angles when mapping the land.

These are just a few examples of how the tangent function is applied in real-world situations.

**Summary**

In this lesson, we explored the tangent function and specifically calculated the tangent of 85 degrees. The tangent function relates the ratio of the length of the side opposite an angle to the length of the side adjacent to the angle. By substituting the appropriate values into the formula and evaluating the numerical value using a calculator, we determined that the tangent of 85 degrees is approximately 11.4301. We also discussed common mistakes and tips to avoid when working with the tangent function. Finally, we highlighted some real-world applications where the tangent function is widely used.

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