Cos 585 Degrees
The value of cos 585 degrees is -0.7071067. . .. Cos 585 degrees in radians is written as cos (585° × π/180°), i.e., cos (13π/4) or cos (10.210176. . .). In this article, we will discuss the methods to find the value of cos 585 degrees with examples.
- Cos 585°: -0.7071067. . .
- Cos 585° in fraction: -(1/√2)
- Cos (-585 degrees): -0.7071067. . .
- Cos 585° in radians: cos (13π/4) or cos (10.2101761 . . .)
What is the Value of Cos 585 Degrees?
The value of cos 585 degrees in decimal is -0.707106781. . .. Cos 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 . . .
∴ cos 585° = cos(10.2101) = -(1/√2) or -0.7071067. . .
Explanation:
For cos 585°, the angle 585° > 360°. Given the periodic property of the cosine function, we can represent it as cos(585° mod 360°) = cos(225°). The angle 585°, coterminal to angle 225°, is located in the Third Quadrant(Quadrant III).
Since cosine function is negative in the 3rd quadrant, thus cos 585 degrees value = -(1/√2) or -0.7071067. . .
Similarly, cos 585° can also be written as, cos 585 degrees = (585° + n × 360°), n ∈ Z.
⇒ cos 585° = cos 945° = cos 1305°, and so on.
Note: Since, cosine is an even function, the value of cos(-585°) = cos(585°).
Methods to Find Value of Cos 585 Degrees
The cosine function is negative in the 3rd quadrant. The value of cos 585° is given as -0.70710. . .. We can find the value of cos 585 degrees by:
- Using Trigonometric Functions
- Using Unit Circle
Cos 585° in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the cos 585 degrees as:
- ± √(1-sin²(585°))
- ± 1/√(1 + tan²(585°))
- ± cot 585°/√(1 + cot²(585°))
- ±√(cosec²(585°) - 1)/cosec 585°
- 1/sec 585°
Note: Since 585° lies in the 3rd Quadrant, the final value of cos 585° will be negative.
We can use trigonometric identities to represent cos 585° as,
- -cos(180° - 585°) = -cos(-405°)
- -cos(180° + 585°) = -cos 765°
- sin(90° + 585°) = sin 675°
- sin(90° - 585°) = sin(-495°)
Cos 585 Degrees Using Unit Circle
To find the value of cos 585 degrees using the unit circle, represent 585° in the form (1 × 360°) + 225° [∵ 585°>360°] ∵ cosine is a periodic function, cos 585° = cos 225°.
- Rotate ‘r’ anticlockwise to form 225° or 585° angle with the positive x-axis.
- The cos of 585 degrees equals the x-coordinate(-0.7071) of the point of intersection (-0.7071, -0.7071) of unit circle and r.
Hence the value of cos 585° = x = -0.7071 (approx)
☛ Also Check:
FAQs on Cos 585 Degrees
What is Cos 585 Degrees?
Cos 585 degrees is the value of cosine trigonometric function for an angle equal to 585 degrees. The value of cos 585° is -(1/√2) or -0.7071 (approx)
What is the Value of Cos 585 Degrees in Terms of Sin 585°?
Using trigonometric identities, we can write cos 585° in terms of sin 585° as, cos(585°) = -√(1 - sin²(585°)). Here, the value of sin 585° is equal to -0.7071.
How to Find Cos 585° in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of cos 585° can be given in terms of other trigonometric functions as:
- ± √(1-sin²(585°))
- ± 1/√(1 + tan²(585°))
- ± cot 585°/√(1 + cot²(585°))
- ± √(cosec²(585°) - 1)/cosec 585°
- 1/sec 585°
☛ Also check: trigonometry table
What is the Value of Cos 585° in Terms of Cosec 585°?
Since the cosine function can be represented using the cosecant function, we can write cos 585° as [√(cosec²(585°) - 1)/cosec 585°]. The value of cosec 585° is equal to -1.41421.
How to Find the Value of Cos 585 Degrees?
The value of cos 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 cos 585° is equal to the x-coordinate (-0.7071). ∴ cos 585° = -0.7071.