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Sin 180 Degrees is equal to zero (0). Sine is one of the three primary trigonometric functions used to calculate the angle or sides of a right-angled triangle.
- Sine of an angle is a trigonometric function denoted by sin x, where x is the angle in consideration.
- Sine Function is the ratio of the perpendicular to the hypotenuse of a right-angled triangle.
- The exact value of Sin 180 Degrees is 0.
- Sin 180 Degrees is written as sin (180° × π/180°) in radians, i.e. sin (π) or sin (3.141592. . .).
Trigonometry is a branch of Mathematics that deals with the length, height, and angles of a right-angled triangle. Trigonometry has a huge number of applications in fields like Science, Technology, and Satellite Navigation.
Read More: NCERT Solutions for Class 11 Mathematics Trigonometric Functions
Key Terms: Sin 180 Degrees, Sine Function, Trigonometric Functions, Right-angled Triangle, Trigonometric Ratio, Hypotenuse, Radians
What is Sine?
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Sine is a primary trigonometric function that is the ratio of the side opposite to an angle and the hypotenuse of the right-angled triangle. It is one of the three main primary trigonometric functions along with Cosine Function (cos), and Tangent Function (tan).
- It is an important periodic function in trigonometry and has a period of 2π.
- It helps to find the value of the sine of the angle between the base and hypotenuse of the right triangle.
- Sine Function is abbreviated as sin x, where x is an acute angle between the base and hypotenuse of the right-angled triangle.
| \(Sin \,x = {Opposite\,Side \over Hypotenuse} = {Perpendicular \over Hypotenuse}\) |
Read More:
| Relevant Concepts | ||
|---|---|---|
| Trigonometry Values | Sin Cos Formulas | Sin 30 Degrees |
| Cos 360 | Sin 90 Degrees | Sin 120 |
Value of Sin 180 Degrees
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Value of Sin 180 Degrees is 0.
| Value of Sin 180 Degrees = 0 |
Sin 180 Degrees may alternatively be represented using the equivalent of the given angle (180 Degrees) in radians (3.14159 . . .). Sin 180 Degrees can be converted into radians using the degrees to radian conversion,
θ in Radians = θ in Degrees × (pi/180°)
180 Degrees = 180° × (π/180°) rad = π or 3.1415 . . .
Thus,
| Sin 180° = Sin (3.1415) = 0 |
Explanation for Value of Sin 180 Degrees
The angle 180° is on the negative y-axis for Sin 180 Degrees. Thus, the value of sin 180° is 0. Since the sine function is a periodic function, we may express sin 180° as,
Sin 180 Degrees = Sin(180° + n × 360°), n ∈ Z
Sin 180° = Sin 540° = Sin 900°...
Note: Since Sine is an odd function, the value of sin (-180°) is = sin(180°).
Trigonometric Functions Detailed Video Explanation
Derivation of Value of Sin 180 Degrees
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Value of Sin 180 Degrees can be derived using two main methods as follows:
- Using Unit Circle
- Using Trigonometric Functions
Both methods are explained in detail below.
Sin 180 Degrees Using Unit Circle
To find the value of Sin 180 Degrees using a unit circle, the following steps need to be followed:
- Rotate a point ‘r’ in an anticlockwise direction to make a 180° angle with the positive x-axis.
- The y-coordinate (0) of the point of intersection (-1, 0) of the unit circle and r is equal to the sin of 180°.
Thus, the value of Sin 180 Degrees is
| Sin 180° = y = 0 |
Sin 180 Degrees Using Trigonometric Functions
Sin 180° Degrees can also be derived using other Trigonometric Functions as follows:
- ± √(1-cos2(180°))
- ± tan 180°/√(1 + tan2(180°))
- ± 1/√(1 + cot2(180°))
- ± √(sec2(180°) - 1)/sec 180°
- 1/cosec 180°
Note: Since 180° is on the negative x-axis, the ultimate value of sin 180° is 0.
To express Sin 180°, several trigonometric identities can be used such as
- sin(180° - 180°) Equals sin (0°).
- -sin(180° + 180°) equals -sin 360°
- cos(90° - 180°) equals cos(-90°).
- -cos(90° + 180°) equals -cos 270°
Read More: Trigonometric Functions Important Questions
Sin 180 Degrees and Sin 0
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Sin 180° and Sin 0° have the same value, which is zero. It is because 180° is in the second quadrant and the value of sine theta that exceeds 90° shifts to the cosine function. That is to say,
Sin (90 + X) = cos X
Sin 180 can be written as:
Sin 180 = Sin (90+90) = Cos 90
Since cos 90 = 0, we get
Sin 180 = 0
As we know, sin 0 = 0
Thus, sin 180 = sin 0.
| Sin 180 Degrees = Sin 0 = 0 |
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Sine Table
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The values of the Sine Function for some specific angles such as 0°, 30°, 45°, 60°, 90°, etc are listed below in the Sine Table:
| Sine Degrees | Sine Radians | Value of Sine Function (sin x) |
|---|---|---|
| sin 0° | sin 0 | 0 |
| sin 30° | sin π/6 | 1/2 |
| sin 45° | sin π/4 | 1/√2 |
| sin 60° | sin π/3 | √3/2 |
| sin 90° | sin π/2 | 1 |
| sin 120° | sin 2π/3 | √3/2 |
| sin 150° | sin 5π/6 | 1/2 |
| sin 180° | sin π | 0 |
| sin 270° | sin 3π/2 | -1 |
| sin 360° | sin 2π | 0 |
Trigonometry Table
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Trigonometry Table is used in order to find the values of standard trigonometric angles such as 0°, 30°, 45°, 60° and 90°. The values of the different angles of six trigonometric ratios namely sine, cosine, tangent, cosecant, secant, and cotangent are listed below:
| Trigonometry Table | ||||||||
|---|---|---|---|---|---|---|---|---|
| Angles (In Degrees) | 0° | 30° | 45° | 60° | 90° | 180° | 270° | 360° |
| Angles (In Radians) | 0° | π/6 | π/4 | π/3 | π/2 | π | 3π/2 | 2π |
| sin | 0 | 1/2 | 1/√2 | √3/2 | 1 | 0 | -1 | 0 |
| cos | 1 | √3/2 | 1/√2 | 1/2 | 0 | -1 | 0 | 1 |
| tan | 0 | 1/√3 | 1 | √3 | ∞ | 0 | ∞ | 0 |
| cot | ∞ | √3 | 1 | 1/√3 | 0 | ∞ | 0 | ∞ |
| cosec | ∞ | 2 | √2 | 2/√3 | 1 | ∞ | -1 | ∞ |
| sec | 1 | 2/√3 | √2 | 2 | ∞ | -1 | ∞ | 1 |
Things to Remember
- Sine is one of the basic trigonometric functions used to calculate the angle or sides of a right-angled triangle.
- Sine is the ratio of the perpendicular to the hypotenuse of a right-angled triangle.
- It is abbreviated as Sin x and is a primary trigonometric function along with Cosine and Tangent.
- Sin 180 Degrees has an exact value of Zero (0).
- Value of Sin 180 Degrees and Sin 0° is the same which is zero (0).
- Value of Sin 180 Degrees is derived using the Unit Circle and other Trigonometric Functions.
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Sample Questions
Ques. What is Sine Function? (3 Marks)
Ans. Sine Function is a primary trigonometric function along with Cosine and Tangent in Trigonometry.
- It is the ratio of the perpendicular and the hypotenuse in a right-angled triangle.
- It is abbreviated as sin x, where x refers to an acute angle between the base and hypotenuse of the right triangle.
Sine Formula is given as:
Sin x = Opposite Side/Hypotenuse = Perpendicular/Hypotenuse
Ques. Prove that b2x2 – a2y2 = a2b2, If:
(i) x = a sec θ, y = b tan θ
(ii) x = a cosec θ, y = b cot θ (CBSE 2014) (3 Marks)
Ans. We need to prove that b2x2 – a2y2 = a2b2.
(i) L.H.S. = b2x2 – a2y2
= b2(a sec θ)2 – a2(b tan θ)2
= b2a2 sec θ – a2b2 tan2θ
= b2a2(sec2 θ – tan2 θ)
= b2a2(1) …[\(\because\) sec2θ – tan2 θ = 1
= a2b2 = R.H.S.
Hence Proved.
(ii) L.H.S. = b2x2 – a2y2
= b2(a cosec θ)2 – a2(b cot θ)2
= b2a2cosec2 θ – a2b2 cot2 θ
= b2a2(cosec2θ – cot2 θ)
= b2a2 (1) ..[\(\because\) cosec2 θ – cot2 θ = 1
= a2b2 = R.H.S.
Hence Proved.
Ques. State the real-life applications of Trigonometry. (3 Marks)
Ans. Here are some real-life applications of Trigonometry:
- It is used in measuring the heights of mountains, towers, or buildings.
- It is used in construction sites to measure several parameters.
- It is used to decide the path of an airplane, from landing to take off in the calculation of speed, direction, and slope.
- It is used in vector algebra, finding components of a vector, cross-product, calculus, waves, oscillations, circular motions, and optics.
Ques. Simplify 2 (sin 180°/sin 90°). (3 Marks)
Ans. Value of Sin 180 Degrees and Sin 90 Degrees are:
- Sin 180° = 0
- Sin 90° = 1
2 sin 180°/sin 90° = 2(sin 180°/sin 90°)
2(0) = 0
Thus, 2 (sin 180°/sin 90°) is equal to zero.
Ques. How to define Sin 180 Degrees in relation to other Trigonometric Functions? (3 Marks)
Ans. The value of Sin 180 Degrees may be expressed in terms of other trigonometric functions as follows:
- ± √(1-cos2(180°))
- ± tan 180°/√(1 + tan2(180°))
- ± 1/√(1 + cot2(180°))
- ± √(sec2(180°) - 1)/sec 180°
- 1/cosec 180°
Ques. What is Sin 180 Degrees in terms of Cosec 180 Degrees? (2 Marks)
Ans. Using trigonometric identities, Sine Function can be expressed in terms of the cosecant function as follows:
Sin 180° = 1/cosec(180°)
The value of cosec 180° is equal to -∞.
Ques. How to calculate Sin 180 Degrees? (3 Marks)
Ans. Sine 180 Degrees can be calculated using the Unit Circle method.
- Create an angle of 180° with the x-axis and determine the coordinates of the corresponding point (-1, 0) on the unit circle.
- The y-coordinate is equal to the value of sin 180°.
Thus, the value of Sin 180° = y = 0
Ques. In ΔABC right angled at B, AB = 24 cm, BC = 7 cm. Determine:
(i) sin A, cos A
(ii) sin C, cos C (3 Marks)
Ans. By Pythagoras’ Theorem,
AC2 = AB2 + BC2 = (24)2 + (7)2
= 576 + 49 = 625
⇒ AC = \(\sqrt{625}\) = 25 cm
(i) sin A = \(\frac{BC}{AC} = \frac{7}{25}\), cos A = \(\frac{AB}{AC} = \frac{24}{25}\)
(II) sin C = \(\frac{AB}{AC} = \frac{24}{25}\), cos C = \(\frac{BC}{AC} = \frac{7}{25}\)
Ques. If sin A = 3/4, Calculate cos A and tan A. (3 Marks)
Ans. Given that sin A = \(\frac{3}{4} = \frac{BC}{AC}\)
Let BC = 3k and AC = 4k
Then by Pythagoras’ Theorem,
AB2 = AC2 – BC2
= (4k)2 – (3k)2
=16k2 – 9k2 = 7k2
⇒ AB = k\(\sqrt{7}\)
Thus,
cos A = \(\frac{AB}{AC} = \frac{\sqrt{7}k}{4k} = \frac{\sqrt{7}}{4}\)
tan A = \(\frac{BC}{AB} = \frac{3k}{\sqrt{7}k} = \frac{3}{\sqrt{7}}\)
Ques. If ∠A and ∠B are acute angles such that cos A = cos B, then show that ∠A = ∠B. (3 Marks)
Ans. Since ∠A and ∠B are acute angles,
Then, ∠C = 90°
cos A = cos B
⇒ \(\frac{AC}{AB} = \frac {BC}{AB}\)
⇒ AC = BC
∴ ∠A = ∠B [Given: Angles opposite to equal sides are equal]
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