Sin pi/4

The value of sin pi/4 is 0.7071067. . .. Sin pi/4 radians in degrees is written as sin ((π/4) × 180°/π), i.e., sin (45°). In this article, we will discuss the methods to find the value of sin pi/4 with examples.

• Sin pi/4: 1/√2
• Sin pi/4 in decimal: 0.7071067. . .
• Sin (-pi/4): -0.7071067. . . or -(1/√2)
• Sin pi/4 in degrees: sin (45°)

What is the Value of Sin pi/4?

The value of sin pi/4 in decimal is 0.707106781. . .. Sin pi/4 can also be expressed using the equivalent of the given angle (pi/4) in degrees (45°).

We know, using radian to degree conversion, θ in degrees = θ in radians × (180°/pi)
⇒ pi/4 radians = pi/4 × (180°/pi) = 45° or 45 degrees
∴ sin pi/4 = sin π/4 = sin(45°) = 1/√2 or 0.7071067. . .

Explanation:

For sin pi/4, the angle pi/4 lies between 0 and pi/2 (First Quadrant). Since sine function is positive in the first quadrant, thus sin pi/4 value = 1/√2 or 0.7071067. . .
Since the sine function is a periodic function, we can represent sin pi/4 as, sin pi/4 = sin(pi/4 + n × 2pi), n ∈ Z.
⇒ sin pi/4 = sin 9pi/4 = sin 17pi/4 , and so on.
Note: Since, sine is an odd function, the value of sin(-pi/4) = -sin(pi/4).

Methods to Find Value of Sin pi/4

The sine function is positive in the 1st quadrant. The value of sin pi/4 is given as 0.70710. . .. We can find the value of sin pi/4 by:

• Using Unit Circle
• Using Trigonometric Functions

Sin pi/4 Using Unit Circle

To find the value of sin π/4 using the unit circle:

• Rotate ‘r’ anticlockwise to form pi/4 angle with the positive x-axis.
• The sin of pi/4 equals the y-coordinate(0.7071) of the point of intersection (0.7071, 0.7071) of unit circle and r.

Hence the value of sin pi/4 = y = 0.7071 (approx)

Sin pi/4 in Terms of Trigonometric Functions

Using trigonometry formulas, we can represent the sin pi/4 as:

• ± √(1-cos²(pi/4))
• ± tan(pi/4)/√(1 + tan²(pi/4))
• ± 1/√(1 + cot²(pi/4))
• ± √(sec²(pi/4) - 1)/sec(pi/4)
• 1/cosec(pi/4)

Note: Since pi/4 lies in the 1st Quadrant, the final value of sin pi/4 will be positive.

We can use trigonometric identities to represent sin pi/4 as,

• sin(pi - pi/4) = sin 3pi/4
• -sin(pi + pi/4) = -sin 5pi/4
• cos(pi/2 - pi/4) = cos pi/4
• -cos(pi/2 + pi/4) = -cos 3pi/4

☛ Also Check:

• tan 3pi/8
• tan pi/4
• cos 2pi/8
• csc pi/2
• sin pi/4
• cot pi/3

FAQs on Sin pi/4

What is Sin pi/4?

Sin pi/4 is the value of sine trigonometric function for an angle equal to pi/4 radians. The value of sin pi/4 is 1/√2 or 0.7071 (approx).

How to Find Sin pi/4 in Terms of Other Trigonometric Functions?

Using trigonometry formula, the value of sin π/4 can be given in terms of other trigonometric functions as:

• ± √(1-cos²(pi/4))
• ± tan(pi/4)/√(1 + tan²(pi/4))
• ± 1/√(1 + cot²(pi/4))
• ± √(sec²(pi/4) - 1)/sec(pi/4)
• 1/cosec(pi/4)

☛ Also check: trigonometric table

What is the Value of Sin pi/4 in Terms of Cot pi/4?

We can represent the sine function in terms of the cotangent function using trig identities, sin pi/4 can be written as 1/√(1 + cot²(pi/4)). Here, the value of cot pi/4 is equal to 1.

What is the Value of Sin pi/4 in Terms of Cosec pi/4?

Since the cosecant function is the reciprocal of the sine function, we can write sin pi/4 as 1/cosec(pi/4). The value of cosec pi/4 is equal to 1.41421.

How to Find the Value of Sin pi/4?

The value of sin pi/4 can be calculated by constructing an angle of π/4 radians with the x-axis, and then finding the coordinates of the corresponding point (0.7071, 0.7071) on the unit circle. The value of sin pi/4 is equal to the y-coordinate (0.7071). ∴ sin pi/4 = 0.7071.