Cos 690 Degrees
The value of cos 690 degrees is 0.8660254. . .. Cos 690 degrees in radians is written as cos (690° × π/180°), i.e., cos (23π/6) or cos (12.042771. . .). In this article, we will discuss the methods to find the value of cos 690 degrees with examples.
- Cos 690°: 0.8660254. . .
- Cos 690° in fraction: √3/2
- Cos (-690 degrees): 0.8660254. . .
- Cos 690° in radians: cos (23π/6) or cos (12.0427718 . . .)
What is the Value of Cos 690 Degrees?
The value of cos 690 degrees in decimal is 0.866025403. . .. Cos 690 degrees can also be expressed using the equivalent of the given angle (690 degrees) in radians (12.04277 . . .)
We know, using degree to radian conversion, θ in radians = θ in degrees × (pi/180°)
⇒ 690 degrees = 690° × (π/180°) rad = 23π/6 or 12.0427 . . .
∴ cos 690° = cos(12.0427) = √3/2 or 0.8660254. . .
Explanation:
For cos 690°, the angle 690° > 360°. Given the periodic property of the cosine function, we can represent it as cos(690° mod 360°) = cos(330°). The angle 690°, coterminal to angle 330°, is located in the Fourth Quadrant(Quadrant IV).
Since cosine function is positive in the 4th quadrant, thus cos 690 degrees value = √3/2 or 0.8660254. . .
Similarly, cos 690° can also be written as, cos 690 degrees = (690° + n × 360°), n ∈ Z.
⇒ cos 690° = cos 1050° = cos 1410°, and so on.
Note: Since, cosine is an even function, the value of cos(-690°) = cos(690°).
Methods to Find Value of Cos 690 Degrees
The cosine function is positive in the 4th quadrant. The value of cos 690° is given as 0.86602. . .. We can find the value of cos 690 degrees by:
- Using Unit Circle
- Using Trigonometric Functions
Cos 690 Degrees Using Unit Circle
To find the value of cos 690 degrees using the unit circle, represent 690° in the form (1 × 360°) + 330° [∵ 690°>360°] ∵ cosine is a periodic function, cos 690° = cos 330°.
- Rotate ‘r’ anticlockwise to form 330° or 690° angle with the positive x-axis.
- The cos of 690 degrees equals the x-coordinate(0.866) of the point of intersection (0.866, -0.5) of unit circle and r.
Hence the value of cos 690° = x = 0.866 (approx)
Cos 690° in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the cos 690 degrees as:
- ± √(1-sin²(690°))
- ± 1/√(1 + tan²(690°))
- ± cot 690°/√(1 + cot²(690°))
- ±√(cosec²(690°) - 1)/cosec 690°
- 1/sec 690°
Note: Since 690° lies in the 4th Quadrant, the final value of cos 690° will be positive.
We can use trigonometric identities to represent cos 690° as,
- -cos(180° - 690°) = -cos(-510°)
- -cos(180° + 690°) = -cos 870°
- sin(90° + 690°) = sin 780°
- sin(90° - 690°) = sin(-600°)
☛ Also Check:
FAQs on Cos 690 Degrees
What is Cos 690 Degrees?
Cos 690 degrees is the value of cosine trigonometric function for an angle equal to 690 degrees. The value of cos 690° is √3/2 or 0.866 (approx)
What is the Value of Cos 690° in Terms of Cosec 690°?
Since the cosine function can be represented using the cosecant function, we can write cos 690° as -[√(cosec²(690°) - 1)/cosec 690°]. The value of cosec 690° is equal to -2.
What is the Value of Cos 690 Degrees in Terms of Sin 690°?
Using trigonometric identities, we can write cos 690° in terms of sin 690° as, cos(690°) = √(1 - sin²(690°)). Here, the value of sin 690° is equal to -0.5.
How to Find Cos 690° in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of cos 690° can be given in terms of other trigonometric functions as:
- ± √(1-sin²(690°))
- ± 1/√(1 + tan²(690°))
- ± cot 690°/√(1 + cot²(690°))
- ± √(cosec²(690°) - 1)/cosec 690°
- 1/sec 690°
☛ Also check: trigonometry table
How to Find the Value of Cos 690 Degrees?
The value of cos 690 degrees can be calculated by constructing an angle of 690° with the x-axis, and then finding the coordinates of the corresponding point (0.866, -0.5) on the unit circle. The value of cos 690° is equal to the x-coordinate (0.866). ∴ cos 690° = 0.866.