What is the exact value of tan /12?

Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles. One of the key trigonometric functions is the tangent, denoted by tan, which calculates the ratio of the length of the opposite side to the length of the adjacent side of a right triangle. In this article,

Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles. One of the key trigonometric functions is the tangent, denoted by tan, which calculates the ratio of the length of the opposite side to the length of the adjacent side of a right triangle. In this article, we will explore the exact value of the tangent of the angle π/12 (pi/12 radians) and its significance in trigonometry.

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Calculating the exact value of tan π/12

To find the exact value of tan π/12, we need to employ trigonometric identities and special angle values. The angle π/12 is not considered a special angle, so its exact value cannot be determined with ease. However, by using certain trigonometric identities, we can express tan π/12 in terms of other known angles and trigonometric values.

Let’s use the sum angle identity for the tangent function, which states that tan(A + B) = (tan A + tan B) / (1 – tan A tan B). By setting A = π/6 (30 degrees) and B = π/12 (15 degrees), we can rearrange the formula as:

tan π/12 = (tan π/6 + tan π/12) / (1 – tan π/6 tan π/12)

But what are the exact values of tan π/6 and tan π/12? Luckily, the tangent of π/6 is a special angle value, since it corresponds to a 30-degree angle, and its exact value is 1/√3. By substituting this value into the formula, we can simplify it further.

tan π/12 = (1/√3 + tan π/12) / (1 – (1/√3)(tan π/12))

Multiplying the numerator and denominator by √3 to eliminate the square root in the denominator, we get:

√3 tan π/12 = (√3 + √3 tan π/12) / (√3 – tan π/12)

Next, we can cross-multiply and rearrange the terms:

√3 tan π/12 – √3 tan π/12 tan π/12 = √3 + √3 tan π/12

Factoring out √3 tan π/12, we obtain:

(√3 – tan π/12) tan π/12 = √3

Finally, solving for tan π/12:

tan π/12 = √3 / (√3 – tan π/12)

This is the exact value of the tangent of the angle π/12. It may not appear in a simpler form, but it accurately represents the ratio of the lengths of the opposite and adjacent sides in a right triangle with a π/12 angle.

Frequently Asked Questions (FAQs)

1. What is the trigonometric definition of the tangent function?

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

2. Why is the exact value of tan π/12 not easy to determine?

The angle π/12 is not considered a special angle, making it challenging to determine its exact value without special trigonometric tables or identities.

3. Can the value of tan π/12 be approximated?

While the exact value of tan π/12 is not simple, it can be approximated using numerical methods or calculators.

4. What are special angles in trigonometry?

Special angles in trigonometry, such as π/6, π/4, and π/3, have exact and easily calculable trigonometric values.

5. How can I find the tangent of other angles?

The tangent function can be calculated using a scientific calculator or by applying trigonometric identities and special angle values when possible.

6. Is trigonometry only applicable to right triangles?

No, while trigonometry has its roots in right triangles, it has broad applications in various fields and can be extended to non-right triangles as well.

7. What is the range of the tangent function?

The tangent function has a periodic nature and can have any real value as the angle varies, except for certain values where it becomes undefined.

8. Can the tangent function exceed numerical bounds?

Yes, the tangent function can yield very large or infinite values for certain angles, particularly those close to π/2 or 3π/2.

9. What other trigonometric functions are related to the tangent?

The secant (sec), cosecant (csc), cotangent (cot), and arctangent (atan) functions are closely related to the tangent function through trigonometric identities.

10. How are trigonometric functions used in real-life applications?

Trigonometric functions are widely used in fields such as engineering, physics, architecture, navigation, and computer graphics to model and solve real-world problems involving angles and distances.

11. Can I use the tangent function to solve non-trigonometric equations?

While the tangent function is mainly used in trigonometry, it can be occasionally employed in solving other types of equations, particularly transcendental equations.

12. Is there a general formula for calculating the tangent of any angle?

There is no general formula to calculate the tangent of any angle, but various trigonometric identities can help express the tangent in terms of known angles and trigonometric values.

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