Cos 390 Degrees
The value of cos 390 degrees is 0.8660254. . .. Cos 390 degrees in radians is written as cos (390° × π/180°), i.e., cos (13π/6) or cos (6.806784. . .). In this article, we will discuss the methods to find the value of cos 390 degrees with examples.
- Cos 390°: 0.8660254. . .
- Cos 390° in fraction: √3/2
- Cos (-390 degrees): 0.8660254. . .
- Cos 390° in radians: cos (13π/6) or cos (6.8067840 . . .)
What is the Value of Cos 390 Degrees?
The value of cos 390 degrees in decimal is 0.866025403. . .. Cos 390 degrees can also be expressed using the equivalent of the given angle (390 degrees) in radians (6.80678 . . .)
We know, using degree to radian conversion, θ in radians = θ in degrees × (pi/180°)
⇒ 390 degrees = 390° × (π/180°) rad = 13π/6 or 6.8067 . . .
∴ cos 390° = cos(6.8067) = √3/2 or 0.8660254. . .
Explanation:
For cos 390°, the angle 390° > 360°. Given the periodic property of the cosine function, we can represent it as cos(390° mod 360°) = cos(30°). The angle 390°, coterminal to angle 30°, is located in the First Quadrant(Quadrant I).
Since cosine function is positive in the 1st quadrant, thus cos 390 degrees value = √3/2 or 0.8660254. . .
Similarly, cos 390° can also be written as, cos 390 degrees = (390° + n × 360°), n ∈ Z.
⇒ cos 390° = cos 750° = cos 1110°, and so on.
Note: Since, cosine is an even function, the value of cos(-390°) = cos(390°).
Methods to Find Value of Cos 390 Degrees
The cosine function is positive in the 1st quadrant. The value of cos 390° is given as 0.86602. . .. We can find the value of cos 390 degrees by:
- Using Unit Circle
- Using Trigonometric Functions
Cos 390 Degrees Using Unit Circle
To find the value of cos 390 degrees using the unit circle, represent 390° in the form (1 × 360°) + 30° [∵ 390°>360°] ∵ cosine is a periodic function, cos 390° = cos 30°.
- Rotate ‘r’ anticlockwise to form 30° or 390° angle with the positive x-axis.
- The cos of 390 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 390° = x = 0.866 (approx)
Cos 390° in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the cos 390 degrees as:
- ± √(1-sin²(390°))
- ± 1/√(1 + tan²(390°))
- ± cot 390°/√(1 + cot²(390°))
- ±√(cosec²(390°) - 1)/cosec 390°
- 1/sec 390°
Note: Since 390° lies in the 1st Quadrant, the final value of cos 390° will be positive.
We can use trigonometric identities to represent cos 390° as,
- -cos(180° - 390°) = -cos(-210°)
- -cos(180° + 390°) = -cos 570°
- sin(90° + 390°) = sin 480°
- sin(90° - 390°) = sin(-300°)
☛ Also Check:
FAQs on Cos 390 Degrees
What is Cos 390 Degrees?
Cos 390 degrees is the value of cosine trigonometric function for an angle equal to 390 degrees. The value of cos 390° is √3/2 or 0.866 (approx)
How to Find Cos 390° in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of cos 390° can be given in terms of other trigonometric functions as:
- ± √(1-sin²(390°))
- ± 1/√(1 + tan²(390°))
- ± cot 390°/√(1 + cot²(390°))
- ± √(cosec²(390°) - 1)/cosec 390°
- 1/sec 390°
☛ Also check: trigonometry table
What is the Value of Cos 390 Degrees in Terms of Sin 390°?
Using trigonometric identities, we can write cos 390° in terms of sin 390° as, cos(390°) = √(1 - sin²(390°)). Here, the value of sin 390° is equal to 0.5.
What is the Exact Value of cos 390 Degrees?
The exact value of cos 390 degrees can be given accurately up to 8 decimal places as 0.86602540 and √3/2 in fraction.
How to Find the Value of Cos 390 Degrees?
The value of cos 390 degrees can be calculated by constructing an angle of 390° 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 390° is equal to the x-coordinate (0.866). ∴ cos 390° = 0.866.