Sin 40 Degrees
The value of sin 40 degrees is 0.6427876. . .. Sin 40 degrees in radians is written as sin (40° × π/180°), i.e., sin (2π/9) or sin (0.698131. . .). In this article, we will discuss the methods to find the value of sin 40 degrees with examples.
 Sin 40°: 0.6427876. . .
 Sin (40 degrees): 0.6427876. . .
 Sin 40° in radians: sin (2π/9) or sin (0.6981317 . . .)
What is the Value of Sin 40 Degrees?
The value of sin 40 degrees in decimal is 0.642787609. . .. Sin 40 degrees can also be expressed using the equivalent of the given angle (40 degrees) in radians (0.69813 . . .).
We know, using degree to radian conversion, θ in radians = θ in degrees × (pi/180°)
⇒ 40 degrees = 40° × (π/180°) rad = 2π/9 or 0.6981 . . .
∴ sin 40° = sin(0.6981) = 0.6427876. . .
Explanation:
For sin 40 degrees, the angle 40° lies between 0° and 90° (First Quadrant). Since sine function is positive in the first quadrant, thus sin 40° value = 0.6427876. . .
Since the sine function is a periodic function, we can represent sin 40° as, sin 40 degrees = sin(40° + n × 360°), n ∈ Z.
⇒ sin 40° = sin 400° = sin 760°, and so on.
Note: Since, sine is an odd function, the value of sin(40°) = sin(40°).
Methods to Find Value of Sin 40 Degrees
The sine function is positive in the 1st quadrant. The value of sin 40° is given as 0.64278. . .. We can find the value of sin 40 degrees by:
 Using Trigonometric Functions
 Using Unit Circle
Sin 40° in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the sin 40 degrees as:
 ± √(1cos²(40°))
 ± tan 40°/√(1 + tan²(40°))
 ± 1/√(1 + cot²(40°))
 ± √(sec²(40°)  1)/sec 40°
 1/cosec 40°
Note: Since 40° lies in the 1st Quadrant, the final value of sin 40° will be positive.
We can use trigonometric identities to represent sin 40° as,
 sin(180°  40°) = sin 140°
 sin(180° + 40°) = sin 220°
 cos(90°  40°) = cos 50°
 cos(90° + 40°) = cos 130°
Sin 40 Degrees Using Unit Circle
To find the value of sin 40 degrees using the unit circle:
 Rotate ‘r’ anticlockwise to form a 40° angle with the positive xaxis.
 The sin of 40 degrees equals the ycoordinate(0.6428) of the point of intersection (0.766, 0.6428) of unit circle and r.
Hence the value of sin 40° = y = 0.6428 (approx)
☛ Also Check:
Examples Using Sin 40 Degrees

Example 1: Using the value of sin 40°, solve: (1cos²(40°)).
Solution:
We know, (1cos²(40°)) = (sin²(40°)) = 0.4132
⇒ (1cos²(40°)) = 0.4132 
Example 2: Find the value of sin 40° if cosec 40° is 1.5557.
Solution:
Since, sin 40° = 1/csc 40°
⇒ sin 40° = 1/1.5557 = 0.6428 
Example 3: Find the value of 5 sin(40°)/7 cos(50°).
Solution:
Using trigonometric identities, we know, sin(40°) = cos(90°  40°) = cos 50°.
⇒ sin(40°) = cos(50°)
⇒ Value of 5 sin(40°)/7 cos(50°) = 5/7
FAQs on Sin 40 Degrees
What is Sin 40 Degrees?
Sin 40 degrees is the value of sine trigonometric function for an angle equal to 40 degrees. The value of sin 40° is 0.6428 (approx).
How to Find Sin 40° in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of sin 40° can be given in terms of other trigonometric functions as:
 ± √(1cos²(40°))
 ± tan 40°/√(1 + tan²(40°))
 ± 1/√(1 + cot²(40°))
 ± √(sec²(40°)  1)/sec 40°
 1/cosec 40°
☛ Also check: trigonometric table
How to Find the Value of Sin 40 Degrees?
The value of sin 40 degrees can be calculated by constructing an angle of 40° with the xaxis, and then finding the coordinates of the corresponding point (0.766, 0.6428) on the unit circle. The value of sin 40° is equal to the ycoordinate (0.6428). ∴ sin 40° = 0.6428.
What is the Value of Sin 40 Degrees in Terms of Tan 40°?
We know, using trig identities, we can write sin 40° as tan 40°/√(1 + tan²(40°)). Here, the value of tan 40° is equal to 0.839099.
What is the Exact Value of sin 40 Degrees?
The exact value of sin 40 degrees can be given accurately up to 8 decimal places as 0.64278760.
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