Sin 585 Degrees
The value of sin 585 degrees is -0.7071067. . .. Sin 585 degrees in radians is written as sin (585° × π/180°), i.e., sin (13π/4) or sin (10.210176. . .). In this article, we will discuss the methods to find the value of sin 585 degrees with examples.
- Sin 585°: -0.7071067. . .
- Sin 585° in fraction: -(1/√2)
- Sin (-585 degrees): 0.7071067. . .
- Sin 585° in radians: sin (13π/4) or sin (10.2101761 . . .)
What is the Value of Sin 585 Degrees?
The value of sin 585 degrees in decimal is -0.707106781. . .. Sin 585 degrees can also be expressed using the equivalent of the given angle (585 degrees) in radians (10.21017 . . .).
We know, using degree to radian conversion, θ in radians = θ in degrees × (pi/180°)
⇒ 585 degrees = 585° × (π/180°) rad = 13π/4 or 10.2101 . . .
∴ sin 585° = sin(10.2101) = -(1/√2) or -0.7071067. . .
Explanation:
For sin 585°, the angle 585° > 360°. Given the periodic property of the sine function, we can represent it as sin(585° mod 360°) = sin(225°). The angle 585°, coterminal to angle 225°, is located in the Third Quadrant(Quadrant III).
Since sine function is negative in the 3rd quadrant, thus sin 585 degrees value = -(1/√2) or -0.7071067. . .
Similarly, sin 585° can also be written as, sin 585 degrees = (585° + n × 360°), n ∈ Z.
⇒ sin 585° = sin 945° = sin 1305°, and so on.
Note: Since, sine is an odd function, the value of sin(-585°) = -sin(585°).
Methods to Find Value of Sin 585 Degrees
The sine function is negative in the 3rd quadrant. The value of sin 585° is given as -0.70710. . .. We can find the value of sin 585 degrees by:
- Using Unit Circle
- Using Trigonometric Functions
Sin 585 Degrees Using Unit Circle
To find the value of sin 585 degrees using the unit circle, represent 585° in the form (1 × 360°) + 225° [∵ 585°>360°] ∵ sine is a periodic function, sin 585° = sin 225°.
- Rotate ‘r’ anticlockwise to form a 225° or 585° angle with the positive x-axis.
- The sin of 585 degrees 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 585° = y = -0.7071 (approx)
Sin 585° in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the sin 585 degrees as:
- ± √(1-cos²(585°))
- ± tan 585°/√(1 + tan²(585°))
- ± 1/√(1 + cot²(585°))
- ± √(sec²(585°) - 1)/sec 585°
- 1/cosec 585°
Note: Since 585° lies in the 3rd Quadrant, the final value of sin 585° will be negative.
We can use trigonometric identities to represent sin 585° as,
- sin(180° - 585°) = sin(-405°)
- -sin(180° + 585°) = -sin 765°
- cos(90° - 585°) = cos(-495°)
- -cos(90° + 585°) = -cos 675°
☛ Also Check:
FAQs on Sin 585 Degrees
What is Sin 585 Degrees?
Sin 585 degrees is the value of sine trigonometric function for an angle equal to 585 degrees. The value of sin 585° is -(1/√2) or -0.7071 (approx).
How to Find Sin 585° in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of sin 585° can be given in terms of other trigonometric functions as:
- ± √(1-cos²(585°))
- ± tan 585°/√(1 + tan²(585°))
- ± 1/√(1 + cot²(585°))
- ± √(sec²(585°) - 1)/sec 585°
- 1/cosec 585°
☛ Also check: trigonometric table
How to Find the Value of Sin 585 Degrees?
The value of sin 585 degrees can be calculated by constructing an angle of 585° 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 585° is equal to the y-coordinate (-0.7071). ∴ sin 585° = -0.7071.
What is the Value of Sin 585° in Terms of Sec 585°?
Since the sine function can be represented using the secant function, we can write sin 585° as √(sec²(585°) - 1)/sec 585°. The value of sec 585° is equal to -1.414214.
What is the Value of Sin 585 Degrees in Terms of Tan 585°?
We know, using trig identities, we can write sin 585° as -tan 585°/√(1 + tan²(585°)). Here, the value of tan 585° is equal to 1.