Sin 420 Degrees
The value of sin 420 degrees is 0.8660254. . .. Sin 420 degrees in radians is written as sin (420° × π/180°), i.e., sin (7π/3) or sin (7.330382. . .). In this article, we will discuss the methods to find the value of sin 420 degrees with examples.
- Sin 420°: 0.8660254. . .
- Sin 420° in fraction: √3/2
- Sin (-420 degrees): -0.8660254. . .
- Sin 420° in radians: sin (7π/3) or sin (7.3303828 . . .)
What is the Value of Sin 420 Degrees?
The value of sin 420 degrees in decimal is 0.866025403. . .. Sin 420 degrees can also be expressed using the equivalent of the given angle (420 degrees) in radians (7.33038 . . .).
We know, using degree to radian conversion, θ in radians = θ in degrees × (pi/180°)
⇒ 420 degrees = 420° × (π/180°) rad = 7π/3 or 7.3303 . . .
∴ sin 420° = sin(7.3303) = √3/2 or 0.8660254. . .
Explanation:
For sin 420°, the angle 420° > 360°. Given the periodic property of the sine function, we can represent it as sin(420° mod 360°) = sin(60°). The angle 420°, coterminal to angle 60°, is located in the First Quadrant(Quadrant I).
Since sine function is positive in the 1st quadrant, thus sin 420 degrees value = √3/2 or 0.8660254. . .
Similarly, sin 420° can also be written as, sin 420 degrees = (420° + n × 360°), n ∈ Z.
⇒ sin 420° = sin 780° = sin 1140°, and so on.
Note: Since, sine is an odd function, the value of sin(-420°) = -sin(420°).
Methods to Find Value of Sin 420 Degrees
The sine function is positive in the 1st quadrant. The value of sin 420° is given as 0.86602. . .. We can find the value of sin 420 degrees by:
- Using Unit Circle
- Using Trigonometric Functions
Sin 420 Degrees Using Unit Circle
To find the value of sin 420 degrees using the unit circle, represent 420° in the form (1 × 360°) + 60° [∵ 420°>360°] ∵ sine is a periodic function, sin 420° = sin 60°.
- Rotate ‘r’ anticlockwise to form a 60° or 420° angle with the positive x-axis.
- The sin of 420 degrees equals the y-coordinate(0.866) of the point of intersection (0.5, 0.866) of unit circle and r.
Hence the value of sin 420° = y = 0.866 (approx)
Sin 420° in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the sin 420 degrees as:
- ± √(1-cos²(420°))
- ± tan 420°/√(1 + tan²(420°))
- ± 1/√(1 + cot²(420°))
- ± √(sec²(420°) - 1)/sec 420°
- 1/cosec 420°
Note: Since 420° lies in the 1st Quadrant, the final value of sin 420° will be positive.
We can use trigonometric identities to represent sin 420° as,
- sin(180° - 420°) = sin(-240°)
- -sin(180° + 420°) = -sin 600°
- cos(90° - 420°) = cos(-330°)
- -cos(90° + 420°) = -cos 510°
☛ Also Check:
FAQs on Sin 420 Degrees
What is Sin 420 Degrees?
Sin 420 degrees is the value of sine trigonometric function for an angle equal to 420 degrees. The value of sin 420° is √3/2 or 0.866 (approx).
What is the Value of Sin 420° in Terms of Cosec 420°?
Since the cosecant function is the reciprocal of the sine function, we can write sin 420° as 1/cosec(420°). The value of cosec 420° is equal to 1.15470.
How to Find Sin 420° in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of sin 420° can be given in terms of other trigonometric functions as:
- ± √(1-cos²(420°))
- ± tan 420°/√(1 + tan²(420°))
- ± 1/√(1 + cot²(420°))
- ± √(sec²(420°) - 1)/sec 420°
- 1/cosec 420°
☛ Also check: trigonometric table
How to Find the Value of Sin 420 Degrees?
The value of sin 420 degrees can be calculated by constructing an angle of 420° with the x-axis, and then finding the coordinates of the corresponding point (0.5, 0.866) on the unit circle. The value of sin 420° is equal to the y-coordinate (0.866). ∴ sin 420° = 0.866.
What is the Value of Sin 420 Degrees in Terms of Cos 420°?
Using trigonometric identities, we can write sin 420° in terms of cos 420° as, sin(420°) = √(1-cos²(420°)). Here, the value of cos 420° is equal to 0.5.