Value of sin 30° is 1/2. In terms of radian sin 30° is written as sin π/6. Trigonometric functions are very important, for various studies such as it is useful to study Wave motion, Movement of light, the study velocity of harmonic oscillators, and other applications.
Sine function, which is one of the basic trigonometric functions, relates the angle of a right triangle to the ratio of the length of the opposite side to the hypotenuse.
sin 30° = 1/2 = 0.5
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What is the Value of Sin 30 degrees?
The value of sin 30 degrees is found by various methods, including the formula of the trigonometric ratio as we know that sin x = Perpendicular/Hypotenuse. In a right-angled triangle, ABC with angle A be 30°.
- Sin 30° in decimal = 0.5
- Sin 30° in radians = sin (π/6) or sin (0.5235) = 0.5
- Sin 30° in fraction = √3/2
- Sin (-30°) = -0.5
How to Find Value of Sin 30 Degree?
Value of sin 30 degree can be calculate:
- Using Geometry
- Using Trigonometric Functions
Let's discuss these methods in detail as follows.
Value of Sin 30 Degree using Geometry
Consider an equilateral triangle PQR where PQ = QR = RP and ∠P = ∠Q = ∠R = 60°. . . (1)
Draw a perpendicular bisector PS of the line QR from vertex P.
Now, QS = RS, and QS = (1/2) QR = (1/2) PQ . . . (2)
In right angle triangle PQS
sin 30° = Perpendicular / Hypotenuse
here, Perpendicular = QS and Hypotenuse = PQ
Thus, sin 30° = QS / PQ
⇒ sin 30° = (1/2)PQ /PQ
⇒ sin 30° = 1/2
Thus, the value of sin 30° is 1/2.
Value of Sin 30 Degree using Trigonometric Function
Trigonometric functions are also called circular functions or trigonometric ratios. The relationship between angles and sides is represented by these trigonometric functions.
The representation of the value of sin 30° using trigonometric functions are:
- sin 30° = √(1 - cos2 30°) = √[1 - (√3/2)2] = √(1 - 3/4) = √(1/4) = 1/2
Why is the Value of Sin 30 Degree equal to Sin 150 Degree?
Value of sin in the first and second quadrants is positive, whereas it is negative in the third and fourth quadrants.
Value of sin 30° and sin 150° are equal.
sin 30 = sin 150 = 1/2
We know that sin (180 - θ) = sin θ
hence, sin 150° = sin (180 - 30) = sin 30°
similarly, sin 0 = sin 180 = 0
Value of Trigonometric Functions
Various values of trigonometric functions are used to solve complex functions. Basic values of trigonometric functions can be learnt from the tale below:
| Angles (In Degrees) | 0 | 30 | 45 | 60 | 90 | 180 | 270 |
|---|---|---|---|---|---|---|---|
| Angles (In Radians) | 0 | π/6 | π/4 | π/3 | π/2 | π | 3π/2 |
| sin | 0 | 1/2 | 1/√2 | √3/2 | 1 | 0 | -1 |
| cos | 1 | √3/2 | 1/√2 | 1/2 | 0 | -1 | 0 |
| tan | 0 | 1/√3 | 1 | √3 | Not Defined | 0 | Not Defined |
| cot | Not Defined | √3 | 1 | 1/√3 | 0 | Not Defined | 0 |
| cosec | Not Defined | 2 | √2 | 2/√3 | 1 | Not Defined | -1 |
| sec | 1 | 2/√3 | √2 | 2 | Not Defined | −1 | Not Defined |
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Solved Examples on Sin 30 Degree
Example 1: In a right-angled triangle, the side opposite the angle of 30° is 7m. Find the length of the Hypotenuse.
Solution:
Given: Perpendicular = 7m
Sin 30 = 1/2
⇒ P/H = 1/2
⇒ 7/H = 1/2
⇒ H = 7 × 2
⇒ H =14m
Example 2: In a right triangle hypotenuse is 20cm, and one side is 10√3cm, find the angles of the triangle.
Solution:
Given: H=20, and B = 10√3
Finding third side using pythagoras theorem.
⇒ P2 + B2 = H2
⇒ P2 + (10√3)2 = 202
⇒ p2 + 300 = 400
⇒ P2 = 100
⇒ P = 10
The third side is 10cm. The ratio of the third side and the hypotenuse is 1/2 (10/20) So there must be an angle of 30° in triangle Since the triangle is a right angle, so the third angle is
90° - 30° = 60°
The Angles of a triangle are 30°, 60°, 90°.
Example 3: In a right-angled triangle adjacent side is 12 cm and the measure of an angle is 60°. Now find the value of the hypotenuse of the triangle.
Solution:
Given adjacent side = 12 cm
tan 60 = Opposite side/Adjacent side = opposite side/12 (tan 60 = √3)
Now,
1 = Opposite side/12
Opposite side = 12√3 cm
Now,
sin A = Opposite side/Hypotenuse
⇒ sin 60 = 10/Hypotenuse
⇒ Hypotenuse = 10/sin 60
⇒ Hypotenuse = 10/(√3/2)
⇒ Hypotenuse = 20√3/3