# How to use the TAN function

**What is the TAN function?**

The TAN function calculates the tangent of an angle in radians.

## 1. Introduction

**What is the trigonometric tangent?**

The tan is a trigonometric function that relates an angle Î¸ in a right triangle to the ratio of the side lengths of a right triangle.

tan(Î¸) = opposite / adjacent

TANGENT Î¸ is equal to the ratio between sides opposite and adjacent, this is only true if the triangle is a right-angled triangle.

**What is a right triangle?**

A right triangle must have one angle that measures exactly 90Â° or Ï€/2 radians. The 90-degree angle is typically represented by a small square or a box in the corner of the angle.

This notation is used to indicate that the angle is a right angle which is Ï€/2 radians or 90 degrees. The image above shows a small black square in the lower right corner of the right triangle.

**What is the angle Î¸?**

The Greek letter theta (Î¸) is commonly used to represent an unknown angle in a right triangle.

**What is the connection between a circle with radius r and a right triangle?**

A circle with radius r is closely related to a right triangle because you can inscribe a right triangle within the circle. In fact, if you draw a radius from the center of the circle to the circle's edge, and then draw a perpendicular line from the edge to the x-axis, you'll create a right triangle with the radius as the hypotenuse.

This connection is fundamental in trigonometry, as it allows you to define the sine, cosine, and tangent functions using the ratios of the sides of the right triangle.

**What is a radian?**

r = radius

Î¸ = 1 radian

arc length = radius

Radians are a fundamental unit of measurement for angles. To be more specific, one radian is equivalent to the angle at the center of a circle that subtends an arc with a length equal to the radius of that circle. In other words, an angle of 1 radian has an arc length equal to the circle's radius.

A circle is 2Ï€ radians or 360 degrees. This means that if you were to measure the angle of a full circle in radians you would get 2Ï€ radians.

On the other hand, a half circle, which is 180 degrees, is equivalent to Ï€ (pi) radians. If you measure the angle of a half circle in radians you get Ï€ radians.

**What is the relationship between the number pi and radians?**

Radians measure angles by the length of the arc they make in a circle rather than degrees. The full circumference of any circle is 2Ï€ multiplied by the circle's radius (2Ï€r).

Since the circumference goes all the way around a circle, that means the full circle measures 2Ï€ radians. Half a circle would be Ï€ radians (half of 2Ï€). A quarter circle is 2Ï€/4 = Ï€/2 radians. An eighth of a circle is 2Ï€/8 = Ï€/4 radians.

Excel has a function that returns the number pi:Â PI function

**What is an arc?**

An arc is a curved segment of a circle's circumference, it is a portion of the circle's curve, defined by two endpoints.

In other words, an arc is formed by two radii intersecting the circumference and the enclosed edge between them.

**What is a circle's radius?**

The radius of a circle is the distance from the center point to any point on the circle's edge or circumference.

**What is radii?**

The plural form of the word "radius".

**What is the Pythagorean theorem?**

The Pythagorean theorem is a mathematical relationship between the sides of a right triangle. In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

a^{2} + b^{2} = c^{2}

a and b are the lengths of the legs of the triangle

c is the length of the hypotenuse

For example, in a triangle with legs 5 and 2:

âˆš(5^{2} + 2^{2)} = âˆš(25 + 4) = âˆš29

The hypotenuse is âˆš29

**What are the main trigonometric functions?**

Function |
Domain (input) |
Range (output) |

sin(x) | All real numbers | (-1, 1) |

cos(x) | All real numbers | (-1, 1) |

tan(x) | All real numbers except multiples of Ï€/2 | (-âˆž, âˆž) |

sec(x) | All real numbers except multiples of Ï€ | (1, âˆž) U (-âˆž, -1) |

csc(x) | All real numbers except integer multiples of Ï€ | (-âˆž, -1) U (1, âˆž) |

cot(x) | All real numbers except integer multiples of Ï€ | (-âˆž, âˆž) |

## 2. Syntax

TAN(*number*)

number |
Required. The radian angle you want to know the tangent of. |

## 3. Example 1

*Calculate the tangent of **Ï€/4 radians in a right triangle**?*

What we know:

- Angle
**A**is Ï€/4 which is 45 degrees - tan A = a/b
- The triangle is a right triangle

Formula in cell C3:

The formula in cell C20 returns 1 which represents the tangent in a right triangle based on an angle of Ï€/4 or 45 degrees. The tangent is the ratio between the opposite side (a) and the adjacent side (b).

*Determine the opposite (a) if the adjacent side (b) is equal to 1 unit and the angle (A) is equal to **Ï€/4 **radians?*

Tangent A = a/b

Tangent Ï€/4 = 1

1 = a/1

If the ratio 1 is equal to a/b then 1 = a/1

a = 1*1

a = 1

The opposite side (a) is 1 unit if angle A is Ï€/4 (45 degrees) and the adjacent side (b) is 1 unit.

## 4. Example 2

*In a right triangle, if one acute angle measures 30 degrees and the opposite is 10 units long, find the length of the adjacent side?*

What we know:

- The adjacent side (b) is unknown.
- Angle A is 30 degrees. We need to convert degrees to radians.
- The opposite side (a) is equal to 10 units.
- TAN A = a/b (opposite side / adjacent side)

Adjacent side (b) = a / SEC A

Formula in cell C20:

The result in cell C20Â is 17.3205080756888 Here is how it is calculated:

- Convert degrees to radians. RADIANS(30) equals 0.523598775598299
- Calculate the tangent. TAN(RADIANS(30) equals 0.577350269189626
- Divide the opposite side to the tangent to get the length of the adjacent side (b).

10/TAN(RADIANS(30)) equals 17.3205080756888

The RADIANS function converts degrees to radians.

## 5. Example 3

*Calculate the tangent ratio if the adjacent side is 3 units and the hypotenuse is 4 units in a right triangle?*

What we know:

- Right triangle
- The adjacent side (b) is 3.
- The hypotenuse (c) is 5.
- The Pythagorean theorem allows us to calculate the opposite side (a).

a^{2}+ b^{2}= c^{2}

a = âˆš(c^{2}- b^{2}) - The tangent ratio is opposite side (a)/ adjacent side (b). TAN A = a/b

Formula in cell C18:

The result is 4/3 which represents the tangent ratioÂ opposite side (a)/ adjacent side (b). Here is how it is calculated:

- Square the hypotenuse . 5^2 = 25
- Square the adjacent side . 3^2 = 9
- Add the squared values. 25 - 9 = 16
- Calculate the square root of the sum. 16^0.5 = 4

We now know that the opposite is 4. All sides in the right triangle is now known. The tangent ratio is opposite side/ adjacent side which becomes 4 / 3. Cell C20 displays 4/3 in decimal form.

### Functions in 'Math and trigonometry' category

The TAN function function is one of 61 functions in the 'Math and trigonometry' category.

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