Sin 7pi/4
The value of sin 7pi/4 is -0.7071067. . .. Sin 7pi/4 radians in degrees is written as sin ((7π/4) × 180°/π), i.e., sin (315°). In this article, we will discuss the methods to find the value of sin 7pi/4 with examples.
- Sin 7pi/4: -(1/√2)
- Sin 7pi/4 in decimal: -0.7071067. . .
- Sin (-7pi/4): 0.7071067. . . or 1/√2
- Sin 7pi/4 in degrees: sin (315°)
What is the Value of Sin 7pi/4?
The value of sin 7pi/4 in decimal is -0.707106781. . .. Sin 7pi/4 can also be expressed using the equivalent of the given angle (7pi/4) in degrees (315°).
We know, using radian to degree conversion, θ in degrees = θ in radians × (180°/pi)
⇒ 7pi/4 radians = 7pi/4 × (180°/pi) = 315° or 315 degrees
∴ sin 7pi/4 = sin 7π/4 = sin(315°) = -(1/√2) or -0.7071067. . .
Explanation:
For sin 7pi/4, the angle 7pi/4 lies between 3pi/2 and 2pi (Fourth Quadrant). Since sine function is negative in the fourth quadrant, thus sin 7pi/4 value = -(1/√2) or -0.7071067. . .
Since the sine function is a periodic function, we can represent sin 7pi/4 as, sin 7pi/4 = sin(7pi/4 + n × 2pi), n ∈ Z.
⇒ sin 7pi/4 = sin 15pi/4 = sin 23pi/4 , and so on.
Note: Since, sine is an odd function, the value of sin(-7pi/4) = -sin(7pi/4).
Methods to Find Value of Sin 7pi/4
The sine function is negative in the 4th quadrant. The value of sin 7pi/4 is given as -0.70710. . .. We can find the value of sin 7pi/4 by:
- Using Trigonometric Functions
- Using Unit Circle
Sin 7pi/4 in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the sin 7pi/4 as:
- ± √(1-cos²(7pi/4))
- ± tan(7pi/4)/√(1 + tan²(7pi/4))
- ± 1/√(1 + cot²(7pi/4))
- ± √(sec²(7pi/4) - 1)/sec(7pi/4)
- 1/cosec(7pi/4)
Note: Since 7pi/4 lies in the 4th Quadrant, the final value of sin 7pi/4 will be negative.
We can use trigonometric identities to represent sin 7pi/4 as,
- sin(pi - 7pi/4) = sin(-3pi/4)
- -sin(pi + 7pi/4) = -sin 11pi/4
- cos(pi/2 - 7pi/4) = cos(-5pi/4)
- -cos(pi/2 + 7pi/4) = -cos 9pi/4
Sin 7pi/4 Using Unit Circle
To find the value of sin 7π/4 using the unit circle:
- Rotate ‘r’ anticlockwise to form 7pi/4 angle with the positive x-axis.
- The sin of 7pi/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 7pi/4 = y = -0.7071 (approx)
☛ Also Check:
FAQs on Sin 7pi/4
What is Sin 7pi/4?
Sin 7pi/4 is the value of sine trigonometric function for an angle equal to 7pi/4 radians. The value of sin 7pi/4 is -(1/√2) or -0.7071 (approx).
How to Find Sin 7pi/4 in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of sin 7π/4 can be given in terms of other trigonometric functions as:
- ± √(1-cos²(7pi/4))
- ± tan(7pi/4)/√(1 + tan²(7pi/4))
- ± 1/√(1 + cot²(7pi/4))
- ± √(sec²(7pi/4) - 1)/sec(7pi/4)
- 1/cosec(7pi/4)
☛ Also check: trigonometry table
What is the Exact Value of sin 7pi/4?
The exact value of sin 7pi/4 can be given accurately up to 8 decimal places as -0.70710678 and -(1/√2) in fraction.
How to Find the Value of Sin 7pi/4?
The value of sin 7pi/4 can be calculated by constructing an angle of 7π/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 7pi/4 is equal to the y-coordinate (-0.7071). ∴ sin 7pi/4 = -0.7071.
What is the Value of Sin 7pi/4 in Terms of Cot 7pi/4?
We can represent the sine function in terms of the cotangent function using trig identities, sin 7pi/4 can be written as -1/√(1 + cot²(7pi/4)). Here, the value of cot 7pi/4 is equal to -1.