Value of Sin 15
Introduction to Trigonometry :
Trigonometry is a branch of Mathematics that deals with the study of sides and angles of a triangle. Trigonometric ratios give the relation between the sides and angles of a right triangle. In a right triangle, the side opposite to the right angle is the longest side which is called the hypotenuse. The other two sides are base and perpendicular with respect to the reference angle considered. The concept of the Pythagoras Theorem is very important for the better understanding of trigonometry. Trigonometry is used in measuring heights and distances in constructions, navigation, and other real-life applications.
Trigonometric Ratios: (image will be updated soon)
There are six trigonometric ratios. The six trigonometric ratios are defined as follows:
Sine of an angle θ,
Sin θ = Opposite / Hypotenuse
Cosine of an angle θ,
Cos θ = Adjacent / Hypotenuse
Tangent of an angle θ,
Tan θ = Opposite / Adjacent
Cotangent of an angle θ,
Cot θ = Adjacent / Opposite
Secant of an angle θ,
Sec θ = Hypotenuse / Adjacent
Cosecant of an angle θ,
Cosec θ = Hypotenuse / Opposite
Trigonometric Ratios of Angle 150:
The six trigonometric functions are sine, cosine, tangent, cotangent, secant, and cosecant of any angle. To calculate the trigonometric ratios of 150, the knowledge of trigonometric ratios of standard angles and half angles is mandatory. Also, one should have knowledge of a few important trigonometric formulas.
The sine of an angle is the ratio of the length of the side opposite to the angle called as opposite side or perpendicular to the length of the side opposite to the right angle in a right triangle which is the longest side called the hypotenuse. To find the sin 15 degrees, the sine and cosine values of standard angles are important.
Derivation of the Value of Sin 15 Degrees:
To find the Sin 15 values, the concept of half angles is used. 150 can be written as 300/2.
Consider an angle ‘P’,
(Sin P/2 + Cos P/2)2 = Sin2 (P/2) + Cos2 (P/2) + 2 Sin (P/2) Cos (P/2)
According to trigonometric formulas, Sin2 A + Cos2 A = 1. Substituting this value in the above equation,
(Sin P/2 + Cos P/2)2 = 1 + 2 Sin (P/2) Cos (P/2)
From the sum to product trigonometric formula,
Sin A + Sin B = 2 Sin (C + D)/2 . Cos (C + D)/2
Using this equation, 2 Sin P/2 Cos P/2 = Sin P/2 + Sin P/2 = Sin P
So the equation can be rewritten as:
(Sin P/2 + Cos P/2)2 = 1 + Sin P
Sin P/2 + Cos P/2 = √(1 + Sin P) → (1)
(Sin P/2 – Cos P/2)2 = Sin2 P/2 + Cos2 P/2 – 2 Sin P/2 Cos P/2
(Sin P/2 – Cos P/2)2 = 1 – Sin P
Sin P/2 – Cos P/2 = √(1 – Sin P) → (2)
Let us consider the value of P = 300. Substituting the value of P in equation 1 and 2,
=> Sin (300/2) + Cos (300/2) = √(1 + Sin 300)
Sin 150 + Cos 150 = √(1 + ½)
Sin 150 + Cos 150 = √3/√2 → (3)
=> Sin (300/2) – Cos (300/2) = √(1 – Sin 300)
Sin 150 – Cos 150 = √(1 – ½)
Sin 150 – Cos 150 = 1/√2 → (4)
Adding Equations (3) and (4),
Sin 150 + Cos 150 + Sin 150 – Cos 150 = √3/√2 – 1/√2
2 Sin 150 = (√3 – 1)/√2
Sin 150 = (√3 – 1)/2√2
Value of Sin 15 degree = (√3 – 1) / 2√2
Alternative Method to Determine Sin 15 Values:
The angle 150 can be written as the difference of 450 and 300.
Sin 150 can be written as Sin (450 – 300). The value can be determined by using the difference formula:
Sin (A – B) = Sin A Cos B – Cos A Sin B
Sin (450 – 300) = Sin 450 Cos 300 – Cos 450 Sin 300
= 1/√2 x √3/2 – 1/√2 x 1/2
= √3/2√2 – 1/2√2
= (√3 – 1) / 2√2
Other trigonometric ratios of 150 can be determined using the relation of other trigonometric ratios with sine of angle 150.
- Using the above-mentioned derivation steps with a little modification, trigonometric ratios of 750 can be calculated.
1. How are the Other 5 Trigonometric Ratios in Terms of Sine of an Angle?
Sine of an angle is defined as the ratio of the side opposite to the angle to the hypotenuse in a right triangle.
Sin θ = Opposite / Hypotenuse
Cosine of an angle can be expressed in terms of sine of an angle as:
Cos θ = √(1 – Sin2θ)
Tangent of an angle can be expressed in terms of its sine as:
Tan θ = Sin θ / Cos θ = Sin θ / √(1 – Sin2θ)
Cotangent of an angle can be expressed in terms of its sine as:
Cot θ = 1/ Tan θ = √(1 – Sin2θ) / Sin θ
Secant of an angle θ can be expressed in terms of its sine as:
Sec θ = 1/ Cos θ = 1/√(1 – Sin2θ)
Cosecant of an angle θ is the reciprocal of its sine.
Cosec θ = 1/ Sin θ
2. What are the Sin 15 Values?
The value of sin 150 is the ratio of the difference between the square root of 3 and 1 to the product of the number 2 and the square root of 2.
There are two methods of determining the sine of angle 150. In one of the methods, sine of the angle 150 can be calculated using the sum and difference formulas in trigonometry. Sum to product formula in trigonometry is also required to determine the sine of 150.
In the alternative method, angle 150 is expressed as the sum of 300 and 450. The difference formula of sine of two individual angles can be used to compute the value of sin 150.
Value of Sin 15 degree = (√3 – 1) / 2√2.