Cos 420 Degrees
The value of cos 420 degrees is 0.5. Cos 420 degrees in radians is written as cos (420° × π/180°), i.e., cos (7π/3) or cos (7.330382. . .). In this article, we will discuss the methods to find the value of cos 420 degrees with examples.
- Cos 420°: 0.5
- Cos 420° in fraction: 1/2
- Cos (-420 degrees): 0.5
- Cos 420° in radians: cos (7π/3) or cos (7.3303828 . . .)
What is the Value of Cos 420 Degrees?
The value of cos 420 degrees in decimal is 0.5. Cos 420 degrees can also be expressed using the equivalent of the given angle (420 degrees) in radians (7.33038 . . .)
We know, using degree to radian conversion, θ in radians = θ in degrees × (pi/180°)
⇒ 420 degrees = 420° × (π/180°) rad = 7π/3 or 7.3303 . . .
∴ cos 420° = cos(7.3303) = 1/2 or 0.5
Explanation:
For cos 420°, the angle 420° > 360°. Given the periodic property of the cosine function, we can represent it as cos(420° mod 360°) = cos(60°). The angle 420°, coterminal to angle 60°, is located in the First Quadrant(Quadrant I).
Since cosine function is positive in the 1st quadrant, thus cos 420 degrees value = 1/2 or 0.5
Similarly, cos 420° can also be written as, cos 420 degrees = (420° + n × 360°), n ∈ Z.
⇒ cos 420° = cos 780° = cos 1140°, and so on.
Note: Since, cosine is an even function, the value of cos(-420°) = cos(420°).
Methods to Find Value of Cos 420 Degrees
The cosine function is positive in the 1st quadrant. The value of cos 420° is given as 0.5. We can find the value of cos 420 degrees by:
- Using Unit Circle
- Using Trigonometric Functions
Cos 420 Degrees Using Unit Circle
To find the value of cos 420 degrees using the unit circle, represent 420° in the form (1 × 360°) + 60° [∵ 420°>360°] ∵ cosine is a periodic function, cos 420° = cos 60°.
- Rotate ‘r’ anticlockwise to form 60° or 420° angle with the positive x-axis.
- The cos of 420 degrees equals the x-coordinate(0.5) of the point of intersection (0.5, 0.866) of unit circle and r.
Hence the value of cos 420° = x = 0.5
Cos 420° in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the cos 420 degrees as:
- ± √(1-sin²(420°))
- ± 1/√(1 + tan²(420°))
- ± cot 420°/√(1 + cot²(420°))
- ±√(cosec²(420°) - 1)/cosec 420°
- 1/sec 420°
Note: Since 420° lies in the 1st Quadrant, the final value of cos 420° will be positive.
We can use trigonometric identities to represent cos 420° as,
- -cos(180° - 420°) = -cos(-240°)
- -cos(180° + 420°) = -cos 600°
- sin(90° + 420°) = sin 510°
- sin(90° - 420°) = sin(-330°)
☛ Also Check:
FAQs on Cos 420 Degrees
What is Cos 420 Degrees?
Cos 420 degrees is the value of cosine trigonometric function for an angle equal to 420 degrees. The value of cos 420° is 1/2 or 0.5
How to Find Cos 420° in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of cos 420° can be given in terms of other trigonometric functions as:
- ± √(1-sin²(420°))
- ± 1/√(1 + tan²(420°))
- ± cot 420°/√(1 + cot²(420°))
- ± √(cosec²(420°) - 1)/cosec 420°
- 1/sec 420°
☛ Also check: trigonometry table
How to Find the Value of Cos 420 Degrees?
The value of cos 420 degrees can be calculated by constructing an angle of 420° with the x-axis, and then finding the coordinates of the corresponding point (0.5, 0.866) on the unit circle. The value of cos 420° is equal to the x-coordinate (0.5). ∴ cos 420° = 0.5.
What is the Value of Cos 420° in Terms of Sec 420°?
Since the secant function is the reciprocal of the cosine function, we can write cos 420° as 1/sec(420°). The value of sec 420° is equal to 2.
What is the Value of Cos 420 Degrees in Terms of Tan 420°?
We know, using trig identities, we can write cos 420° as 1/√(1 + tan²(420°)). Here, the value of tan 420° is equal to 1.732050.