The term Trigonometry is derived from Greek words i.e; trigonon and metron, which implies triangle and to measure respectively, θ. There are 6 Trigonometry ratios namely, Sine, Cosine, Tangent, Cotangent, Secant, and Cosecant. These trigonometry ratios tell the different combinations in a right-angled triangle.

### Trigonometric ratios

Trigonometric ratios are ratios of the lengths of the right-angled triangle. These ratios can be used to determine the ratios of any two sides out of a total of three sides of a right-angled triangle.

**Sine function:**The sine ratio for the given angle θ in a right-angled triangle is defined as the ratio of lengths of its opposite side to its hypotenuse.

i.e., Sinθ = AB/AC

**Cosine function:**The Cosine ratio for the given angle θ in a right-angled triangle is defined as the ratio of lengths of its adjacent side to its hypotenuse.

i.e, Cosθ = BC/AC

**Tangent Function:**The Tangent ratio for the given angle θ in a right-angled triangle is defined as the ratio of lengths of its opposite side to its adjacent.

i.e, Tanθ = AB/BC

**Cotangent Function:**The Cotangent ratio for the given angle θ in a right-angled triangle is defined as the ratio of lengths of its adjacent side to its opposite. It’s the reciprocal of the tan ratio.

i.e, Cotθ = BC/AB =1/Tanθ

**Secant Function:**The Secant ratio for the given angle θ in a right-angled triangle is defined as the ratio of lengths of its Hypotenuse side to its adjacent.

i.e, Secθ = AC/BC

**Cosecant Function:**The Cosecant ratio for the given angle θ in a right-angled triangle is defined as the ratio of lengths of its hypotenuse side to its opposite.

i.e, Cosecθ = AC/AB

### Sin Theta Formula

In a Right-angled triangle, the sine function or sine theta is defined as the ratio of the opposite side to the hypotenuse of the triangle. In a triangle, the Sine rule helps to relate the sides and angles of the triangle with its circumradius(R) i.e, a/SinA = b/SinB = c/SinC = 2R. Where a, b, and c are lengths of the triangle, and A, B, C are angles, and R is circumradius.

Sin θ = (opposite side / hypotenuse)

From the above figure, sine θ can be written as

Sinθ = AB / AC

According to the Pythagoras theorem, we know that AB

^{2 }+ BC^{2 }= AC^{2}. On dividing both sides by AC^{2}⇒ (AB/AC)

^{2 }+ (BC/AC)^{2}⇒ Sin

^{2}θ + Cos^{2}θ = 1

### Sample Problems

**Question 1: If the sides of the right-angled triangle △ABC which is right-angled at B are 7, 25, and 24 respectively. Then find the value of SinC?**

**Solution:**

As we know that Sinθ = (Opposite side/hypotenuse)

SinC = 24/25

**Question 2: If two sides of a right-angled triangle are 3 and 5 then find the sine of the smallest angle of the triangle?**

**Solution:**

By Pythagoras theorem, other side of the triangle is found to be 4.

As the smaller side lies opposite to the smaller angle,

Then Sine of smaller angle is equal to 3/5.

**Question 3: If sinA = 12/13 in the triangle △ABC, then find the least possible lengths of sides of the triangle?**

**Solution:**

As we know, Sinθ = opposite/hypotenuse

Here, opposite side = 12 and hypotenuse = 13

Then by the pythagoras theorem, other side of the triangle is 5 units

**Question 4: If the lengths of sides of a right-angled △PQR are in A.P. then find the sine values of the smaller angles?**

**Solution: **

The only possible Pythagorean triplet for the given condition is (3, 4, 5).

Therefore, the sine values of the smaller sides are 3/5 and 4/5

**Question 5: In a triangle △XYZ if CosX=1/2 then find the value of SinY?**

**Solution:**

From the given data, the angle X is equal to 60 degrees, then Y=30 degrees as it’s a right angled triangle.

Therefore, SinY = Sin30°

Y = 1/2

**Question 6: If sinθ.Secθ = 1/5 then find the value of Sinθ?**

**Solution:**

As secθ = 1/cosθ

Secθ = Tanθ = 1/5.

Therefore, opposite side= k and adjacent side is 5k and hypotenuse = √26 k.

Then Sinθ = k/√26 k

= 1/√26

**Question 7: In a right-angled triangle, if the ratio of smaller angles is 1:2 then find the sum of sines of smaller angles of the triangle?**

**Solution:**

Let the smaller angles be A, B. As A:B = 1:2.

So, A = k and B = 2k. As A + B = 90.

⇒ k + 2k = 90

⇒ k = 30.

Therefore, the other angles are 30 and 60

So, their sine values are 1/2 and √3/2

Therefore, the sum of the sines is (1 + √3)/2

## Sin x Formula:

It is one of the formulae of the Sin function from the six other trigonometric functions.

## Sin angle Questions

**Example:** If cos x = 4/5, Find the value of Sin x?

**Solution:** Using Trigonometric identities: Sin^{2}x = 1- Cos^{2}x

Sin^{2}x = 1 – (4/5)^{2}

= 1 – 16/25

= (25 – 16) / 25

= 9/25

Sin x =

= 3/5

**Example 2**: If Cosec x = 4/7, find the Sin x?

**Solution**: As Cosec x = 1/sin x

= 1/ 4/7

= 7/4

## Sin Theta

The sine function ‘or’ Sin Theta is one of the three most common trigonometric functions along with cosine and tangent. In right-angled trigonometry, the sine function is defined as the ratio of the opposite side and hypotenuse.

The mathematical denotation of the sine function is,

## More About Sin Theta

Sin theta formula can also be calculated from the product of the tangent of the angle with the cosine of the angle.

sin(θ)=tan(θ)×cos(θ)

The derivative of sin(θ) in calculus is cos(θ) and the integral of it is −cos(θ). The reciprocal of sin theta is cosec(θ).

Below is a table of sin theta values for different degrees and radians.

## Important Sin Theta Formula

Some important properties of the sine function and sin theta formula are:

**SIN θ Values **

The below table shows the values of sin θ for various degrees

Sin θ degree | value |

0 | 0 |

30 | ½ |

45 | 1√2 |

60 | 3√2 |

90 | 1 |

180 | 0 |

**More Sin Theta Formula**

- Sin (- D) = – sin D
- Sin (90 – R) = cos R
- Sin (180 – B) = sin B
- sin2 M + cos2 M = 1
- A+B =Sin A ×Cos B+Cos A ×Sin B
- A-B =Sin A ×Cos B-Cos A ×Sin B
- Sin 2 θ = 2 sin θ. cos θ
- Sin 3 θ = 3 sin θ – 4 sin3 θ

**Solved Examples**

**1. If cos θ = 24/25 then find the value of sin ****θ using sin theta formula.**

We know that, sin² θ + cos² θ = 1

Sin²θ=1- cos²

**2. On a building site, John was working. He’s trying to get to the top of the wall. A 44-foot ladder connects the top of the wall to a location on the ground. The ladder forms a 60-degree angle with the ground. What would the wall’s height be?**

We know that sin 60 = √3/2, also sin θ here in this case would be wall’s height/ladder

**3. In a 7,24,25 right angled triangle at B. what would be the value of Sin C ?**

Sin C = Opposite side/hypotenuse

= AB/AC

=24/25

**4. In a right angled triangle △ABC at B, if CosA=1/2 then find the value of SinC?**

Since the given triangle is a right-angled triangle, and we know that cos 60 = ½ then angle C will measure 30 as per the definition of triangles.

Well then sin 30 = ½

**Example on Sin x Formula**

**1. If Cos x = ****3****5****, then find the value of Sin x.**

Then Find the Value of Sin x.

**Solution: **We know that, cos θ = BaseHypotenuse

On comparing the given ratio, Base = 3, Hypotenuse= 5.

Now we also know Pythagoras theorem, which says,

(Hypotenuse)² = (Base)² + (Perpendicular)²

⇒ (Perpendicular)² = (Hypotenuse)² – (Base)²

⇒ (Perpendicular)² = 5² – 3²

⇒ (Perpendicular)² = 25 – 9

⇒ (Perpendicular)² = 16

⇒ (Perpendicular)² = 4

Here we are considering only positive signs because the length of the side can’t be negative.

Sin x – 45

Hence this is the required answer.

**2. If Cosec x = ****6****7**** **then Find the Value of Sin x.

**Solution:** We know that sin x = 1cosec x

Using the above value we have, sin x = 16/7

Hence, sin x = 76

## What Is the Sin Formula?

The sine of an angle of a right-angled triangle is the ratio of its perpendicular (that is opposite to the angle) to the hypotenuse. The sin formula is given as:

- sin θ = Perpendicular / Hypotenuse.
- sin(θ + 2nπ) = sin θ for every θ
- sin(−θ) = − sin θ

## Solved Examples Using Sin Formula

**Example 1: **Find the value of sin780^{o}.

**Solution**

To find: The value of sin 780^{o }using the sin formula.

We have:

780^{o }= 720^{o }+ 60^{o}

⇒780^{o }= 60^{o}

⇒sin(780^{o}) = sin(60^{o}) = √3/2

**Answer: **The value of sin780^{o} is √3/2.

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