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
There are six trigonometric ratios.
The six trigonometric ratios are defined as follows:
Sine of an angle θ,
sin θ = \[\frac{Opposite}{Hypotenuse}\]
Cosine of an angle θ,
cos θ = \[\frac{Adjacent}{Hypotenuse}\]
Tangent of an angle θ,
tan θ = \[\frac{Opposite}{Adjacent}\]
Cotangent of an angle θ,
cot θ = \[\frac{Adjacent}{Opposite}\]
Secant of an angle θ,
sec θ = \[\frac{Hypotenuse}{Adjacent}\]
Cosecant of an angle θ,
cosec θ = \[\frac{Hypotenuse}{Opposite}\]
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 degree value, the concept of half angles is used.
150 can be written as \[\frac{300}{2}\].
Consider an angle ‘P’, (sin \[\frac{P}{2}\] + cos \[\frac{P}{2}\])2 = sin2 (\[\frac{P}{2}\]) + cos2 (\[\frac{P}{2}\]) + 2 sin (\[\frac{P}{2}\]) cos (\[\frac{P}{2}\])
According to trigonometric formulas, sin² A + cos² A = 1.
Substituting this value in the above equation,
(sin \[\frac{P}{2}\] + cos \[\frac{P}{2}\])² = 1 + 2 sin (\[\frac{P}{2}\]) cos (\[\frac{P}{2}\])
From the sum to product trigonometric formula, sin A + sin B = \[\frac{2 sin (C + D)}{2}\] . \[\frac{cos (C + D)}{2}\]
Using this equation, 2 sin \[\frac{P}{2}\] cos \[\frac{P}{2}\] = sin \[\frac{P}{2}\] + sin \[\frac{P}{2}\] = sin P
So the equation can be rewritten as:
(sin \[\frac{P}{2}\] + cos \[\frac{P}{2}\])2 = 1 + sin P sin \[\frac{P}{2}\] + cos \[\frac{P}{2}\] = \[\sqrt{(1 + sin P)}\] → (1)
(sin \[\frac{P}{2}\] - cos \[\frac{P}{2}\])² = sin² \[\frac{P}{2}\] + cos² \[\frac{P}{2}\] - 2 sin \[\frac{P}{2}\] cos \[\frac{P}{2}\]
(Sin \[\frac{P}{2}\] - Cos \[\frac{P}{2}\])²= 1 - sin P
(sin \[\frac{P}{2}\] - cos \[\frac{P}{2}\]) = \[\sqrt{(1 - sin P)}\] → (2)
Let us consider the value of P = 300.
Substituting the value of P in equation 1 and 2,
=> sin (\[\frac{300}{2}\]) + cos (\[\frac{300}{2}\]) = \[\sqrt{(1 + sin 300)}\] sin 150 + cos 150 = \[\sqrt{(1 + \frac{1}{2})}\] sin 150 + cos 150 = \[\frac{\sqrt{3}}{\sqrt{2}}\] → (3)
=> sin (\[\frac{300}{2}\]) - cos (\[\frac{300}{2}\]) = \[\sqrt{(1 - sin 300)}\] sin 150 - cos 150 = \[\sqrt{(1 - \frac{1}{2})}\] sin 150 - cos 150 = \[\frac{1}{\sqrt{2}}\] → (4)
Adding Equations (3) and (4),
sin 150 + cos 150 + sin 150 - cos 150 = \[\frac{\sqrt{3}}{\sqrt{2}}\] - 1/√22 sin 150 = (√3 - 1)/√2sin 150 = \[\frac{(\sqrt{3-1})}{2\sqrt{2}}\]
Value of sin 15 degree = \[\frac{(\sqrt{3-1})}{2\sqrt{2}}\]
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
= \[\frac{1}{\sqrt{2}}\] x \[\frac{\sqrt{3}}{2}\] - \[\frac{1}{\sqrt{2}}\] x \[\frac{1}{2}\] = \[\frac{\sqrt{3}}{2\sqrt{2}}\] - \[\frac{1}{2\sqrt{2}}\]
= \[\frac{(\sqrt{3-1})}{2\sqrt{2}}\]
Fun Facts:
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.
Simpler Method for finding the Value of Sin 15°
The simplest method for finding the value of sin 15° is given here:
sin 15° = Sin (45 – 30)°
Sin (45-30)° = Sin 45° cos 30° – cos 45° sin 30°
Sin 15° = (\[\frac{1}{\sqrt{2}}\])(\[\frac{\sqrt{3}}{2}\])-(\[\frac{1}{\sqrt{2}}\])(\[\frac{1}{2}\])
sin 15° = (\[\frac{\sqrt{3}}{2\sqrt{2}}\])-(\[\frac{1}{2\sqrt{2}}\])
sin 15° = \[\frac{(\sqrt{3-1})}{2\sqrt{2}}\]
Value of Sin 15 in Decimal Form
As we know, sin 15° = \[\frac{(\sqrt{3-1})}{2\sqrt{2}}\].
Now, we will rationalize the denominator on the right hand side of the above expression.
= \[\frac{(\sqrt{3-1})}{2\sqrt{2}}\]
Multiply and divide √2.
⇒ \[\frac{(\sqrt{3-1})}{2\sqrt{2}}\] x \[(\frac{\sqrt{2}}{\sqrt{2}})\]
⇒ \[\frac{\sqrt{2}(\sqrt{3-1})}{4}\]
⇒ \[\frac{(\sqrt{6}-\sqrt{2})}{4}\]
⇒ 0.2588 (up to four decimal places)
Thus, the value of sin 15° is equal to = 0.2588
FAQs on Value of Sin 15
1. What is the value of sin 150 in terms of Trigonometric Functions?
The value of sin 150 in terms of trigonometric functions can be represented in the following way:
± √(1-cos²(15°))
± tan 15°/√(1 + tan²(15°))
± 1/√(1 + cot²(15°))
± √(sec²(15°) - 1)/sec 15°
1/cosec 15°
The value of sin 15° will be positive because 15° lies in the first quadrant. The trigonometric identities can be used to represent sin 15° as follows:
sin(180° - 15°) = sin 165°
-sin(180° + 15°) = -sin 195°
cos(90° - 15°) = cos 75°
-cos(90° + 15°) = -cos 105°
2. How will you find the value of sin 15°?
The value of sin 15° can be found by making an angle of 15° with the x-axis and then finding the coordinates of the corresponding point (0.9659, 0.2588) on the unit circle. The value of sin 15° is equal to the y-coordinate (0.2588). Thus, sin 15° = 0.2588.
3. What is the value of sin 60° + sin 15°?
You know that
Sin 60° = √3/2
sin 15° = (√3–1)/2√2
Now, on adding both values we will get;
Sin 60° + sin 15° = √3/2 + [(√3–1)/2√2]
= [2√3 + √6 – √2]/4
4. 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
5. 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, the 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.
