# Value of Tan 15

Tan 15 degrees using trigonometry formulas will be evaluated here in this article. But first, let us get a brief about trigonometry concepts. As we know, trigonometry is a branch which deals with angles and lengths of a right triangle. It states the relationships that involve lengths and the angles of triangles. In other words, trigonometry is the study of the right triangle and is derived from the Greek word, where, ‘TRI’ means three, ‘GON’ means sides and ‘METRON’ means ways to measure.

In Trigonometry, Sine, Cosine and Tangent are the three primary ratios. Based on these trigonometric ratios, the whole trigonometric functions, identities and formulas are designed. To find the tangent value of 15 degrees, we are going to use one of these trigonometric formulae. Also, we will find out the value of tan 15, with the help of the other two ratios, which is sine and cosine, and also with the help of trigonometry table.

Tan (15°) can be found if we know the value of sin 15 degrees and cos 15 degrees. The tangent of an angle is equal to the ratio of sine and cosine functions of the same angle, in the right angle triangle. Therefore, once we have found the values of sin 15 degrees and cos 15 degrees, we can easily find the corresponding tangent value. Now let us learn how to find the value of tan 15, in this article.

## What is the Value of Tan 15 Degrees?

Before we try to find the tan 15 degrees value, let’s have a look at trigonometry table for sin, cos and tan.

Angle |
0° |
30° |
45^{°} |
60° |
90° |

Sin θ |
0 | 1/2 | 1/√2 | √3/2 | 1 |

Cos θ |
1 | √3/2 | 1/√2 | 1/2 | 0 |

Tan θ |
0 | 1/√3 | 1 | √3 | ∞ |

From the above table, we have the values of tan, sin and cos ratios for 0°, 30°, 45°, 60° and 90°. Now, by using these values, we have to find the value of tan (15°). Let’s get started,

Tan 15°^{ }= Tan(45 – 30)°

By the trigonometry formula, we know,

Tan (A – B) = (Tan A – Tan B) /(1 + Tan A Tan B)

Therefore, we can write,

tan(45 – 30)° = tan 45° – tan 30°/1+tan 45° tan 30°

Now putting the values of tan 45° and tan 30° from the table we get;

tan(45 – 30)° = (1 – 1/√3)/ (1 + 1.1/√3)

tan (15°) = √3 – 1/ √3 + 1

Hence, **the value of **tan (15°)** is √3 – 1/√3 + 1. **

We can further resolve the above-resulted expression by putting the value of √3, which is equal to 1.732.

∴ Tan (15°) = (1.732 – 1)/(1.732 + 1) = 0.2679

Or tan (15°)** ≈ 0.27**

### Tan 15° With Respect To sin and cos function

Similarly, we can also find the value of tangent 15 degrees, by knowing the value of sin 15 and cos 15 degrees.

Tan (15°) = sin 15/cos 15

Tan 15° = sin 15/cos 15

Sin 15° = sin (45 – 30)° and cos 15 = cos (45 – 30)°

∴ tan (15°) = sin (45 – 30)° /cos (45 – 30)°

From the trigonometry formulas, we know,

sin(A – B) = sin A cos B – cos A sin B

and cos (A – B) = cos A cos B + sin A sin B

Therefore,

tan (15°)= (sin 45° cos 30° – cos 45° sin 30°)/ (cos 45° cos 30° + sin 45° sin 30°)

Putting the values of sin 30°, sin 45°, cos 30° and cos 45°, we get,

tan 15° = [(1/√2).(√3/2) – (1/√2).(½)] / [(1/√2).(√3/2) + (1/√2).(½)]

Solving the above equation we have,

**tan 15° = √3 – 1/ √3 + 1**

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