Problem 12 - Algebra 2 and Trigonometry

Question

In \(3-44,\) find the exact value. $$ \tan 60^{\circ} $$

Step-by-Step Solution

Answer

The exact value of \( \tan 60^{\circ} \) is \( \sqrt{3} \).

Step 1: Recall the Tangent Function for Special Angles

The tangent of an angle in a right triangle is defined as the ratio of the opposite side to the adjacent side. For special angles like \( 60^{\circ} \), it's helpful to remember the values of trigonometric functions. You can often find these in trigonometric tables or recall them from special triangles like the 30-60-90 triangle.

Step 2: Identify the 30-60-90 Triangle

In the 30-60-90 triangle, the sides have a specific ratio of 1:√3:2. For a \( 60^{\circ} \) angle, the side opposite to it is \( \sqrt{3} \) and the adjacent side is 1.

Step 3: Calculate Tangent for 60°

The formula for tangent is \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \). For \( \tan 60^{\circ} \), substitute the known values: \( \tan 60^{\circ} = \frac{\sqrt{3}}{1} \).

Step 4: Simplify the Expression

Simplifying \( \frac{\sqrt{3}}{1} \) gives \( \sqrt{3} \) since dividing by 1 does not change the value.

Key Concepts

Special Angles

In the study of trigonometric functions, there are a few angles, known as special angles, that often appear and hold key significance. These angles include 30°, 45°, and 60°. They are called special because their trigonometric function values—sine, cosine, and tangent—are exact and frequently used.
Understanding these angles can make calculations much easier, especially without a calculator. For instance:

• For 30°, \( an 30° = \frac{1}{\sqrt{3}}\)
• For 45°, \( an 45° = 1\)
• For 60°, \( an 60° = \sqrt{3}\)

These values can be directly obtained from special triangles or unit circle properties. It's important to remember these since they simplify many mathematical problems. While learning, keep a mental or physical chart handy with these angles to speed up the problem-solving process.

30-60-90 Triangle

One of the most fundamental triangles you'll encounter in trigonometry is the 30-60-90 triangle. This triangle is always a right triangle, which means one of its angles is 90°. The other two angles are, as the name suggests, 30° and 60°.
The side lengths of a 30-60-90 triangle have a consistent ratio of 1:√3:2. This means if the shortest side (opposite the 30° angle) is 1 unit, then the length of the side opposite the 60° angle is √3, and the hypotenuse is 2.

• 30° angle: shortest side, 1
• 60° angle: middle length side, \(\sqrt{3}\)
• Hypotenuse opposite 90°: longest side, 2

This predictable ratio aids in quickly determining side lengths and angles in many geometric problems. Memorizing this triangle helps not only with mental math but also builds a strong foundation for solving more complex trigonometric problems.

Tangent Function

The tangent function is one of three foundational trigonometric functions alongside sine and cosine. It relates the angles in a right triangle to ratios of two of its sides. Specifically, \( an \theta\) is defined as the ratio of the opposite to the adjacent side for a given angle \( \theta \) in a right triangle.

• The formula: \( \tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} \)
• For example, \( \tan 60° = \frac{\sqrt{3}}{1} = \sqrt{3} \)

This function is extremely useful because it allows you to find an angle's tangent value if you know the side lengths. This is particularly helpful for special angles, where these values can be found directly without calculations. By using the properties of the 30-60-90 triangle, knowing that the side opposite 60° is √3 and the side adjacent is 1, we can quickly compute \( \tan 60° = \sqrt{3} \). Understanding tangent and its relationship with these triangle properties simplifies evaluating trigonometric functions significantly.