What is the Value of tan 30?

S6-SA2-0043

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

The value of tan 30 refers to the tangent of an angle measuring 30 degrees in a right-angled triangle. Tangent is a trigonometric ratio that relates the length of the side opposite to an angle to the length of the side adjacent to that angle.

Simple Example

Quick Example

Imagine a ladder leaning against a wall, making a 30-degree angle with the ground. If you want to find the ratio of the height the ladder reaches on the wall to the distance of the ladder's base from the wall, you would use tan 30. This ratio is a fixed number, no matter how long the ladder is, as long as the angle is 30 degrees.

Worked Example

Step-by-Step

Let's find the value of tan 30 using an equilateral triangle.

1. Start with an equilateral triangle ABC with side length '2a'. All angles are 60 degrees.
---2. Draw an altitude AD from vertex A to side BC. This altitude bisects BC and angle A.
---3. Now, in the right-angled triangle ADB, angle B is 60 degrees, angle BAD is 30 degrees, and angle ADB is 90 degrees.
---4. The side BD is half of BC, so BD = a. The hypotenuse AB = 2a.
---5. Using the Pythagorean theorem in triangle ADB, AD^2 + BD^2 = AB^2. So, AD^2 + a^2 = (2a)^2.
---6. AD^2 + a^2 = 4a^2, which means AD^2 = 3a^2. Therefore, AD = sqrt(3)a.
---7. In triangle ADB, for angle 30 degrees (angle BAD), the opposite side is BD = a, and the adjacent side is AD = sqrt(3)a.
---8. tan 30 = Opposite / Adjacent = BD / AD = a / (sqrt(3)a) = 1 / sqrt(3).

Answer: The value of tan 30 is 1/sqrt(3).

Why It Matters

Understanding tan 30 helps engineers design structures like bridges and buildings, ensuring stability. It's used in physics to calculate forces and trajectories, crucial for rockets in space technology. Even in AI/ML, these concepts form the basis for understanding spatial relationships in computer vision.

Common Mistakes

MISTAKE: Confusing tan 30 with sin 30 or cos 30. | CORRECTION: Remember the SOH CAH TOA mnemonic: SOH (Sin = Opposite/Hypotenuse), CAH (Cos = Adjacent/Hypotenuse), TOA (Tan = Opposite/Adjacent).

MISTAKE: Forgetting to rationalize the denominator, leaving the answer as 1/sqrt(3). | CORRECTION: Always rationalize the denominator by multiplying both numerator and denominator by sqrt(3), so 1/sqrt(3) becomes sqrt(3)/3.

MISTAKE: Not knowing the side lengths for standard angles like 30, 45, 60 degrees. | CORRECTION: Memorize the values for these common angles, or learn how to quickly derive them using a 30-60-90 triangle or a 45-45-90 triangle.

Practice Questions

Try It Yourself

QUESTION: What is the reciprocal of tan 30? | ANSWER: cot 30, which is sqrt(3).

QUESTION: If tan x = 1/sqrt(3), what is the value of x? | ANSWER: x = 30 degrees.

QUESTION: A tower casts a shadow 30 meters long when the sun's elevation angle is 30 degrees. What is the height of the tower? (Hint: The angle of elevation is the angle formed with the ground.) | ANSWER: Height = 30 * tan 30 = 30 * (1/sqrt(3)) = 30/sqrt(3) = 10*sqrt(3) meters.

MCQ

Quick Quiz

Which of the following is the correct value of tan 30?

sqrt(3)

2026-01-02T00:00:00.000Z

sqrt(3)/2

1/sqrt(3)

The Correct Answer Is:

tan 30 is defined as the ratio of the opposite side to the adjacent side in a 30-degree right triangle, which is 1/sqrt(3). Options A, B, C are values for other trigonometric ratios or angles.

Real World Connection

In the Real World

When civil engineers plan a new flyover or a ramp for a building in cities like Mumbai or Delhi, they often use trigonometry to calculate the required slope. If a ramp needs to rise at a specific angle like 30 degrees, knowing tan 30 helps them determine the relationship between the ramp's height and its horizontal length, ensuring it's safe and accessible.

Key Vocabulary

Key Terms

Trigonometry: The branch of mathematics dealing with the relations of sides and angles of triangles. | Tangent: A trigonometric ratio of the opposite side to the adjacent side in a right-angled triangle. | Right-angled triangle: A triangle with one angle measuring 90 degrees. | Angle of Elevation: The angle between the horizontal line and the line of sight to an object above the horizontal. | Rationalize: The process of removing a radical from the denominator of a fraction.

What's Next

What to Learn Next

Great job understanding tan 30! Next, explore the values of tan 45 and tan 60. Learning these standard angle values will build a strong foundation for solving more complex problems involving heights, distances, and angles in real-world situations.