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Sin 3pi/4

The value of sin 3pi/4 is 0.7071067. . .. Sin 3pi/4 radians in degrees is written as sin ((3π/4) × 180°/π), i.e., sin (135°). In this article, we will discuss the methods to find the value of sin 3pi/4 with examples.

• Sin 3pi/4: 1/√2
• Sin 3pi/4 in decimal: 0.7071067. . .
• Sin (-3pi/4): -0.7071067. . . or -(1/√2)
• Sin 3pi/4 in degrees: sin (135°)

## What is the Value of Sin 3pi/4?

The value of sin 3pi/4 in decimal is 0.707106781. . .. Sin 3pi/4 can also be expressed using the equivalent of the given angle (3pi/4) in degrees (135°).

We know, using radian to degree conversion, θ in degrees = θ in radians × (180°/pi)
⇒ 3pi/4 radians = 3pi/4 × (180°/pi) = 135° or 135 degrees
∴ sin 3pi/4 = sin 3π/4 = sin(135°) = 1/√2 or 0.7071067. . .

Explanation:

For sin 3pi/4, the angle 3pi/4 lies between pi/2 and pi (Second Quadrant). Since sine function is positive in the second quadrant, thus sin 3pi/4 value = 1/√2 or 0.7071067. . .
Since the sine function is a periodic function, we can represent sin 3pi/4 as, sin 3pi/4 = sin(3pi/4 + n × 2pi), n ∈ Z.
⇒ sin 3pi/4 = sin 11pi/4 = sin 19pi/4 , and so on.
Note: Since, sine is an odd function, the value of sin(-3pi/4) = -sin(3pi/4).

## Methods to Find Value of Sin 3pi/4

The sine function is positive in the 2nd quadrant. The value of sin 3pi/4 is given as 0.70710. . .. We can find the value of sin 3pi/4 by:

• Using Trigonometric Functions
• Using Unit Circle

## Sin 3pi/4 in Terms of Trigonometric Functions

Using trigonometry formulas, we can represent the sin 3pi/4 as:

• ± √(1-cos²(3pi/4))
• ± tan(3pi/4)/√(1 + tan²(3pi/4))
• ± 1/√(1 + cot²(3pi/4))
• ± √(sec²(3pi/4) – 1)/sec(3pi/4)
• 1/cosec(3pi/4)

Note: Since 3pi/4 lies in the 2nd Quadrant, the final value of sin 3pi/4 will be positive.

We can use trigonometric identities to represent sin 3pi/4 as,

• sin(pi – 3pi/4) = sin pi/4
• -sin(pi + 3pi/4) = -sin 7pi/4
• cos(pi/2 – 3pi/4) = cos(-pi/4)
• -cos(pi/2 + 3pi/4) = -cos 5pi/4

## Sin 3pi/4 Using Unit Circle

To find the value of sin 3π/4 using the unit circle:

• Rotate ‘r’ anticlockwise to form 3pi/4 angle with the positive x-axis.
• The sin of 3pi/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 3pi/4 = y = 0.7071 (approx)

☛ Also Check:

## Examples Using Sin 3pi/4

1. Example 1: Find the value of 2 × (sin(3pi/8) cos(3pi/8)). [Hint: Use sin 3pi/4 = 0.7071]

Solution:

Using the sin 2a formula,

2 sin(3pi/8) cos(3pi/8) = sin(2 × 3pi/8) = sin 3pi/4
∵ sin 3pi/4 = 0.7071
⇒ 2 × (sin(3pi/8) cos(3pi/8)) = 0.7071

2. Example 2: Find the value of 5 sin(3pi/4)/7 cos(-pi/4).

Solution:

Using trigonometric identities, we know, sin(3pi/4) = cos(pi/2 – 3pi/4) = cos(-pi/4).

⇒ sin(3pi/4) = cos(-pi/4)
⇒ Value of 5 sin(3pi/4)/7 cos(-pi/4) = 5/7

3. Example 3: Using the value of sin 3pi/4, solve: (1-cos²(3pi/4)).

Solution:

We know, (1-cos²(3pi/4)) = (sin²(3pi/4)) = 0.5
⇒ (1-cos²(3pi/4)) = 0.5

## FAQs on Sin 3pi/4

### What is Sin 3pi/4?

Sin 3pi/4 is the value of sine trigonometric function for an angle equal to 3pi/4 radians. The value of sin 3pi/4 is 1/√2 or 0.7071 (approx).

### How to Find the Value of Sin 3pi/4?

The value of sin 3pi/4 can be calculated by constructing an angle of 3π/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 3pi/4 is equal to the y-coordinate (0.7071). ∴ sin 3pi/4 = 0.7071.

What is the Value of Sin 3pi/4 in Terms of Cosec 3pi/4?

Since the cosecant function is the reciprocal of the sine function, we can write sin 3pi/4 as 1/cosec(3pi/4). The value of cosec 3pi/4 is equal to 1.41421.

How to Find Sin 3pi/4 in Terms of Other Trigonometric Functions?

Using trigonometry formula, the value of sin 3π/4 can be given in terms of other trigonometric functions as:

• ± √(1-cos²(3pi/4))
• ± tan(3pi/4)/√(1 + tan²(3pi/4))
• ± 1/√(1 + cot²(3pi/4))
• ± √(sec²(3pi/4) – 1)/sec(3pi/4)
• 1/cosec(3pi/4)

☛ Also check: trigonometric table

What is the Value of Sin 3pi/4 in Terms of Cos 3pi/4?

Using trigonometric identities, we can write sin 3pi/4 in terms of cos 3pi/4 as, sin(3pi/4) = √(1-cos²(3pi/4)). Here, the value of cos 3pi/4 is equal to −(1/√2) .

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