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.
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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 AB2 + BC2 = AC2. On dividing both sides by AC2
⇒ (AB/AC)2 + (BC/AC)2
⇒ Sin2θ + Cos2θ = 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: Sin2x = 1- Cos2x
Sin2x = 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 = 35, 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 = 67 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 sin780o.
Solution
To find: The value of sin 780o using the sin formula.
We have:
780o = 720o + 60o
⇒780o = 60o
⇒sin(780o) = sin(60o) = √3/2
Answer: The value of sin780o is √3/2.
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