What is the exact value of tan 30?

**What is the exact value of tan 30?**

The exact value of tan 30 is 0.577.

Trigonometry is a branch of mathematics that deals with the relationships between angles and the sides of triangles. In trigonometry, the tangent function (tan) is used to determine the ratio of the length of the side opposite an angle to the length of the side adjacent to the angle. The value of the tangent function varies depending on the angle given. In the case of 30 degrees, the exact value of the tangent function can be calculated.

To find the exact value of tan 30, we should first understand the concept of a special triangle called the 30-60-90 triangle. This triangle has angles measuring 30 degrees, 60 degrees, and 90 degrees. In this triangle, the side lengths follow a specific ratio. The shorter leg, opposite the 30-degree angle, is half the length of the hypotenuse, and the longer leg, opposite the 60-degree angle, is multiplied by the square root of 3.

By using the properties of the 30-60-90 triangle, we can determine the exact value of tan 30. Since the tangent function is defined as the ratio of the opposite side (in this case, the shorter leg) to the adjacent side (the longer leg), we need to take the ratio of these two lengths. Given that the shorter leg is half the length of the hypotenuse, and the longer leg is the square root of 3 times the length of the shorter leg, we can calculate:

tan 30 = Opposite / Adjacent
= (1/2) / (√3 * (1/2))
= 1 / (√3/2)
= 2 / √3

To rationalize the denominator, we multiply both the numerator and the denominator by √3:

tan 30 = (2 / √3) * (√3 / √3)
= 2√3 / 3

Therefore, the exact value of tan 30 is 2√3 / 3, which is approximately 0.577. This means that the ratio of the length of the side opposite a 30-degree angle to the length of the side adjacent to it is approximately 0.577.

FAQs:

1. How is the tangent function defined?

The tangent function is defined as the ratio of the opposite side to the adjacent side of a right triangle.

2. What is a special triangle?

A special triangle is a triangle with angles that have easy-to-work-with ratios for their sides. Examples include the 30-60-90 and 45-45-90 triangles.

3. How can the value of tan 30 be determined using the 30-60-90 triangle?

By understanding the ratios of the sides in a 30-60-90 triangle, we can calculate the value of tan 30.

4. Is the value of tan 30 rational or irrational?

The value of tan 30, which is 2√3 / 3, is an irrational number since √3 is irrational.

5. What is the numerical approximation of tan 30?

The numerical approximation of tan 30 is approximately 0.577.

6. What is the relationship between tan 30 and sin 30?

The relationship between tan 30 and sin 30 is that tan 30 is equal to sin 30 divided by cos 30.

7. How can the value of tan 30 be used in real-life situations?

The value of tan 30 can be used in various fields such as engineering, physics, and architecture for calculating angles and distances.

8. What is the range of values for the tangent function?

The tangent function can have values ranging from negative infinity to positive infinity, as it is not limited like sine and cosine.

9. Is the tangent function periodic?

Yes, the tangent function is periodic, with a period of 180 degrees or π radians.

10. Can the tangent function be negative?

Yes, the tangent function can be negative for angles in the second and fourth quadrants.

11. How is the value of tan 30 related to the unit circle?

The value of tan 30 can be obtained by determining the y-coordinate divided by the x-coordinate of the point where the angle 30 degrees intersects the unit circle.

12. Are there alternative methods to find the value of tan 30?

Yes, there are alternative methods such as using trigonometric identities or calculators to find the value of tan 30, but the method using the 30-60-90 triangle is one of the simplest ways.

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