Sin pi/8

The value of sin pi/8 is 0.3826834. . .. Sin pi/8 radians in degrees is written as sin ((π/8) × 180°/π), i.e., sin (22.5°). In this article, we will discuss the methods to find the value of sin pi/8 with examples.

• Sin pi/8 in decimal: 0.3826834. . .
• Sin (-pi/8): -0.3826834. . .
• Sin pi/8 in degrees: sin (22.5°)

What is the Value of Sin pi/8?

The value of sin pi/8 in decimal is 0.382683432. . .. Sin pi/8 can also be expressed using the equivalent of the given angle (pi/8) in degrees (22.5°).

We know, using radian to degree conversion, θ in degrees = θ in radians × (180°/pi)
⇒ pi/8 radians = pi/8 × (180°/pi) = 22.5° or 22.5 degrees
∴ sin pi/8 = sin π/8 = sin(22.5°) = 0.3826834. . .

Explanation:

For sin pi/8, the angle pi/8 lies between 0 and pi/2 (First Quadrant). Since sine function is positive in the first quadrant, thus sin pi/8 value = 0.3826834. . .
Since the sine function is a periodic function, we can represent sin pi/8 as, sin pi/8 = sin(pi/8 + n × 2pi), n ∈ Z.
⇒ sin pi/8 = sin 17pi/8 = sin 33pi/8 , and so on.
Note: Since, sine is an odd function, the value of sin(-pi/8) = -sin(pi/8).

Methods to Find Value of Sin pi/8

The sine function is positive in the 1st quadrant. The value of sin pi/8 is given as 0.38268. . .. We can find the value of sin pi/8 by:

• Using Trigonometric Functions
• Using Unit Circle

Sin pi/8 in Terms of Trigonometric Functions

Using trigonometry formulas, we can represent the sin pi/8 as:

• ± √(1-cos²(pi/8))
• ± tan(pi/8)/√(1 + tan²(pi/8))
• ± 1/√(1 + cot²(pi/8))
• ± √(sec²(pi/8) - 1)/sec(pi/8)
• 1/cosec(pi/8)

Note: Since pi/8 lies in the 1st Quadrant, the final value of sin pi/8 will be positive.

We can use trigonometric identities to represent sin pi/8 as,

• sin(pi - pi/8) = sin 7pi/8
• -sin(pi + pi/8) = -sin 9pi/8
• cos(pi/2 - pi/8) = cos 3pi/8
• -cos(pi/2 + pi/8) = -cos 5pi/8

Sin pi/8 Using Unit Circle

To find the value of sin π/8 using the unit circle:

• Rotate ‘r’ anticlockwise to form pi/8 angle with the positive x-axis.
• The sin of pi/8 equals the y-coordinate(0.3827) of the point of intersection (0.9239, 0.3827) of unit circle and r.

Hence the value of sin pi/8 = y = 0.3827 (approx)

☛ Also Check:

• sin 3pi/4
• cos 2pi/7
• tan 5pi/6
• sin 11pi/6
• sin 2pi
• cot pi/8

FAQs on Sin pi/8

What is Sin pi/8?

Sin pi/8 is the value of sine trigonometric function for an angle equal to pi/8 radians. The value of sin pi/8 is 0.3827 (approx).

What is the Exact Value of sin pi/8?

The exact value of sin pi/8 can be given accurately up to 8 decimal places as 0.38268343.

What is the Value of Sin pi/8 in Terms of Tan pi/8?

We know, using trig identities, we can write sin pi/8 as tan(pi/8)/√(1 + tan²(pi/8)). Here, the value of tan pi/8 is equal to 0.414213.

How to Find Sin pi/8 in Terms of Other Trigonometric Functions?

Using trigonometry formula, the value of sin π/8 can be given in terms of other trigonometric functions as:

• ± √(1-cos²(pi/8))
• ± tan(pi/8)/√(1 + tan²(pi/8))
• ± 1/√(1 + cot²(pi/8))
• ± √(sec²(pi/8) - 1)/sec(pi/8)
• 1/cosec(pi/8)

☛ Also check: trigonometry table

How to Find the Value of Sin pi/8?

The value of sin pi/8 can be calculated by constructing an angle of π/8 radians with the x-axis, and then finding the coordinates of the corresponding point (0.9239, 0.3827) on the unit circle. The value of sin pi/8 is equal to the y-coordinate (0.3827). ∴ sin pi/8 = 0.3827.