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## Sine, Cosine and Tangent

*hree Functions, but same idea.*

## Right Triangle

Sine, Cosine and Tangent are the main functions used in Trigonometry and are based on a Right-Angled Triangle.

Before getting stuck into the functions, it helps to give a **name** to each side of a right triangle:

## Sine, Cosine and Tangent

**Sine**, **Cosine** and **Tangent** (often shortened to **sin**, **cos** and **tan**) are each a **ratio of sides** of a right angled triangle:

**Examples**

(get your calculator out and check them!)

### Example: what are the sine, cosine and tangent of 45° ?

The classic 45° triangle has two sides of 1 and a hypotenuse of √2:

## Why?

Why are these functions important?

- Because they let us work out angles when we know sides
- And they let us work out sides when we know angles

## Less Common Functions

To complete the picture, there are 3 other functions where we divide one side by another, but they are not so commonly used.

They are equal to **1 divided by cos**, **1 divided by sin**, and **1 divided by tan**:

## Sin Cos Tan Values

In trigonometry, sin cos and tan values are the primary functions we consider while solving trigonometric problems. These trigonometry values are used to measure the angles and sides of a right-angle triangle. Apart from sine, cosine and tangent values, the other three major values are cotangent, secant and cosecant.

When we find sin cos and tan values for a triangle, we usually consider these angles: 0°, 30°, 45°, 60° and 90°. It is easy to memorise the values for these certain angles. The trigonometric values are about the knowledge of standard angles for a given triangle as per the trigonometric ratios (sine, cosine, tangent, cotangent, secant and cosecant).

## Sin Cos Tan Formula

The three ratios, i.e. sine, cosine and tangent have their individual formulas. Suppose, ABC is a right triangle, right-angled at B, as shown in the figure below:

Now as per sine, cosine and tangent formulas, we have here:

- Sine θ = Opposite side/Hypotenuse = BC/AC
- Cos θ = Adjacent side/Hypotenuse = AB/AC
- Tan θ = Opposite side/Adjacent side = BC/AB

We can see clearly from the above formulas, that:

Tan θ = sin θ/cos θ

Now, the formulas for other trigonometry ratios are:

- Cot θ = 1/tan θ = Adjacent side/ Side opposite = AB/BC
- Sec θ = 1/Cos θ = Hypotenuse / Adjacent side = AC / AB
- Cosec θ = 1/Sin θ = Hypotenuse / Side opposite = AC / BC

The other side of representation of trigonometric values formulas are:

- Tan θ = sin θ/cos θ
- Cot θ = cos θ/sin θ
- Sin θ = tan θ/sec θ
- Cos θ = sin θ/tan θ
- Sec θ = tan θ/sin θ
- Cosec θ = sec θ/tan θ

## Sin Cos Tan Chart

Let us see the table where the values of sin cos tan sec cosec and tan are provided for the important angles 0°, 30°, 45°, 60° and 90°

Angles (in degrees) | 0° | 30° | 45° | 60° | 90° |

Angles (in radian) | 0 | π/6 | π/4 | π/3 | π/2 |

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 | ∞ |

Cot θ | ∞ | √3 | 1 | 1/√3 | 0 |

Sec θ | 1 | 2/√3 | √2 | 2 | ∞ |

Cosec θ | ∞ | 2 | √2 | 2/√3 | 1 |

## How to find Sin Cos Tan Values?

To remember the trigonometric values given in the above table, follow the below steps:

- First divide the numbers 0,1,2,3, and 4 by 4 and then take the positive roots of all those numbers.
- Hence, we get the values for sine ratios,i.e., 0, ½, 1/√2, √3/2, and 1 for angles 0°, 30°, 45°, 60° and 90°
- Now, write the values of sine degrees in reverse order to get the values of cosine for the same angles.
- As we know, tan is the ratio of sin and cos, such as tan θ = sin θ/cos θ. Thus, we can get the values of tan ratio for the specific angles.

**Sin Values**

sin 0° = √(0/4) = 0

sin 30° = √(1/4) = ½

sin 45° = √(2/4) = 1/√2

sin 60° = √3/4 = √3/2

sin 90° = √(4/4) = 1

**Cos Values**

cos 0° = √(4/4) = 1

cos 30° = √(3/4) = √3/2

cos 45° = √(2/4) = 1/√2

cos 60° = √(1/4) = 1/2

cos 90° = √(0/4) = 0

**Tan Values**

tan 0° = 0/1 = 0

tan 30° = [(√1/4)/√(3/4)] = 1/√3

tan 45° = √(2/4) = 1/√2

tan 60° = [(√3/2)/(½)] = √3

tan 90° = 1/0 = ∞

Hence, the sin cos tan values are found.

## What is Sin Cos Tan in Trigonometry?

Sin, cos, and tan are the three primary trigonometric ratios, namely, sine, cosine, and tangent respectively, where each of which gives the ratio of two sides of a right-angled triangle. We know that the longest side of a right-angled triangle is known as the “hypotenuse” and the other two sides are known as the “legs.” That means, in trigonometry, the longest side of a right-angled triangle is still known as the “hypotenuse” but the other two legs are named to be:

- opposite side and
- adjacent side

We decide the “opposite” and “adjacent” sides based upon the angle which we are talking about.

- The “opposite side” or the perpendicular is the side that is just “opposite” to the angle.
- The “adjacent side” or the base is the side(other than the hypotenuse) that “touches” the angle.

## Sin Cos Tan Formulas

Sin, cos, and tan functions in trigonometry are defined in terms of two of the three sides (opposite, adjacent, and hypotenuse) of a right-angled triangle. Here are the formulas of sin, cos, and tan.

sin θ = Opposite/Hypotenuse

cos θ = Adjacent/Hypotenuse

tan θ = Opposite/Adjacent

Apart from these three trigonometric ratios, we have another three ratios called csc, sec, and cot which are the reciprocals of sin, cos, and tan respectively. Let us understand these sin, cos, and tan formulas using the example given below.

**Example:** Find the sin, cos, and tan of the triangle for the given angle θ.

**Solution:**

In the triangle, the longest side (or) the side opposite to the right angle is the hypotenuse. The side opposite to θ is the opposite side or perpendicular. The side adjacent to θ is the adjacent side or base.

Now we find sin θ, cos θ, and tan θ using the above formulas:

sin θ = Opposite/Hypotenuse = 3/5

cos θ = Adjancent/Hypotenuse = 4/5

tan θ = Opposite/Adjacent = 3/4

**Trick to remember sin cos tan formulas in trigonometry:** Here is a trick to remember the formulas of sin, cos, and tan. We can use the acronym “SOHCAHTOA” as shown below,

### Sin Cos Tan Chart

Sin cos tan chart/table is a chart with the trigonometric values of sine, cosine, and tangent functions for some standard angles 0^{o}, 30^{o}, 45^{o}, 60^{o}, and 90^{o}. We can refer to the trig table given below to directly pick values of sin, cos, and tan values for standard angles.

## Tips to Remember Sin Cos Tan Table

The tips that you need to memorize from this chart are:

- The angles 0
^{o}, 30^{o}, 45^{o}, 60^{o}, and 90^{o }in order. - The first row (of sin) can be remembered like this: 0/2, √1/2, √2/2, √3/2.
- That’s all you need to remember because:

The row of cos is as same as the row of sin just in the reverse order. - Each value in the row of tan is obtained by dividing the corresponding values of sin by cos because tan = sin/cos.

You can see how is tan = sin/cos here:

sin θ/cos θ = (Opposite/Hypotenuse) ÷ (Adjacent/Hypotenuse) = (Opposite/Hypotenuse) × (Hypotenuse/Adjacent) = Opposite/Adjacent = tan θ

## Sin Cos Tan on Unit Circle

The values of sin, cos, and tan can be calculated for any given angle using the unit circle. Unit circle in a coordinate plane is a circle of unit radius of 1, frequently centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane, especially in trigonometry. For any point on unit circle, given with the coordinates(x, y), the sin, cos and tan ratios can be given as,

- sin θ = y/1
- cos θ = x/1
- tan θ = y/x

where θ is the angle the line joining the point and origin forms with the positive x-axis.

## Application of Sin Cos Tan in Real Life

Trigonometry ratios sin, cos, tan find application in finding heights and distances in our daily lives. We use sin, cos, and tan to solve many real-life problems. Here is an example to understand the applications of sin, cos and tan.

**Example:** A ladder leans against a brick wall making an angle of 50^{o} with the horizontal. If the ladder is at a distance of 10 ft from the wall, then up to what height of the wall the ladder reaches?

**Solution:**

Let us assume that the ladder reaches till x ft of the wall.

Using the given information:

Here, we know the adjacent side (which is 10 ft) and we have to find the opposite side (which is x ft). So we use the relation between the opposite and the adjacent sides which is tan.

tan 50^{o} = x/10

x = 10 tan 50^{o}

x ≈ 11.9 ft

Here, tan 50^{o} is calculated using the calculator and the final answer is rounded up to 1 decimal. Therefore, the ladder reaches up to 11.9 ft of the wall.

## Examples on Sin Cos Tan

**Example 1: Find value of cos θ with respect to the triangle, such that the sides opposite and adjacent to θ measure 6 units and 8 units respectively.**

**Solution:**

To find cos θ, we need the adjacent side and the hypotenuse.

Here, the adjacent side = 8.

But we are not given the hypotenuse.

To find this, we use Pythagoras theorem:

hypotenuse^{2} = opposite^{2} + adjacent^{2}

= 6^{2} + 8^{2}

= 100

hypotenuse = √100 = 10

Therefore, cos θ = Adjacent/Hypotenuse = 8/10 = 4/5

cos θ = 4/5

**Example 2: Find the exact length of the shadow cast by a 15 ft lamp post when the angle of elevation of the sun is 60º.**

**Solution:**

Let us assume that the length of the shadow of the lamp post is x ft.

Using the given information:

Applying tan to the given triangle

tan 60^{o} = 15/x

x = 15/tan 60^{o}

x = 15/√3 (Using trigonometry chart)

x = 15√3/3

∴ The length of the shadow of the lamp post is 15√3/3 ft.

**Example 3: Solve the given expression using sin cos tan values: tan 60 ^{o}(sec 60^{o}/cosec 60^{o})**

**Solution:**

We know, sec 60^{o}/cosec 60^{o} = sin 60^{o}/cos 60^{o}

⇒ tan 60^{o}(sec 60^{o}/cosec 60^{o}) = tan 60^{o}(sin 60^{o}/cos 60^{o}) = tan 60^{o} × tan 60^{o}

= (√3)^{2} = 3

**Example 4: If sin θ = 2/3 and tan θ < 0, what is the value of cos θ?**

It is given that sin θ is positive and tan θ is negative. So θ must be in Quadrant II, where cos θ is negative. Now, sin θ = 2/3 = Opposite/Hypotenuse.

So we can assume that Opposite = 2k, Hypotenuse = 3k.

By Pythagoras theorem,

Adjacent^{2} = Hypotenuse^{2} – Opposite^{2}

= (3k)^{2} – (2k)^{2} = 5k^{2}

Adjacent = √5k

Therefore, cos θ = – Adjacent/Hypotenuse = -√5k/3k = -√5/3

## FAQs on Sin Cos Tan

### What is Meant By Sin Cos Tan in Trigonometry?

Sin, cos, and tan are the basic trigonometric ratios in trigonometry, used to study the relationship between the angles and sides of a triangle (especially of a right-angled triangle).

### How to Use Sin Cos Tan?

We can use sin cos and tan to solve real-world problems. To solve any problem, we first draw the figure that describes the problem and we use the respective trigonometric ratio to solve the problem.

### How to Find Sin Cos Tan Values?

To find sin, cos, and tan we use the following formulas:

- sin θ = Opposite/Hypotenuse
- cos θ = Adjacent/Hypotenuse
- tan θ = Opposite/Adjacent

For finding sin, cos, and tan of standard angles, you can use the trigonometry table.

### What is the Table for Sine, Cosine, and Tangent in Trigonometry?

The trigonometry table or chart for sin, cos, and tan are used to find these trigonometric values for standard angles 0^{o}, 30^{o}, 45^{o}, 60^{o}, and 90^{o}. Using the sin cos tan table, we can directly find the sin cos tan values for these angles and use in problems.

### How to Find Value of Tan Using Sin and Cos?

The value of the tan function for any angle θ in terms of sin and cos can be given using the formula, tan θ = sin θ/cos θ.

### What is the Trick to Remember the Formula to Find Sin Cos Tan?

We can remember the sin cos tan formulas using the word “SOHCAHTOA”. This word can be used to remember the ratios of the respective sides involved in the calculation of sin, cos, and tan values in trigonometry as given below,

**S**in θ =**O**pposite/**H**ypotenuse**C**os θ =**A**djacent/**H**ypotenuse**T**an θ =**O**pposite/**A**djacent

### Where is Sin Cos Tan Used?

Sin, cos, and tan find application in trigonometry to calculate the value of these functions while establishing a relationship between the angles and sides of a triangle (especially of a right-angled triangle).

### What is Formula for Sin, Cos, and Tan?

The formulas to find the sin, cos, and tan for any angle θ in any right-angled triangle are given below,

- sin θ = Opposite/Hypotenuse
- cos θ = Adjacent/Hypotenuse
- tan θ = Opposite/Adjacent

✅ Math Formulas ⭐️⭐️⭐️⭐️⭐

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