What is the cosine of 1

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How can we calculate the value of $\cos 1$ where the angle is in radians (and not degrees). If this isn't possible, can we somehow find whether this value would be rational or irrational?

Table of Contents Show

Cos 1 Radian
The Value of the Inverse Cos of 1
The Value of the Inverse Cos of -1
What is the Value of Cos 1 Degrees?
Methods to Find Value of Cos 1 Degrees
Cos 1 Degrees Using Unit Circle
Cos 1° in Terms of Trigonometric Functions
How to Find Cos 1° in Terms of Other Trigonometric Functions?
How to Find the Value of Cos 1 Degrees?
What is the Exact Value of cos 1 Degrees?
What is the Value of Cos 1 Degrees in Terms of Tan 1°?

P.S: I know how to determine the irrationality of $\cos 1$ when angle is in degrees, and also am aware of its explicit formula. But those methods cannot be used here.

Cos-1, COS-1, cos-1, or cos−1 may refer to:

Cos-1, one of two commonly used COS cell lines
cos x−1 = cos(x)−1 = −(1−cos(x)) = −ver(x) or negative versine of x, the additive inverse (or negation) of an old trigonometric function
cos−1y = cos−1(y), sometimes interpreted as arccos(y) or arccosine of y, the compositional inverse of the trigonometric function cosine (see below for ambiguity)
cos−1x = cos−1(x), sometimes interpreted as (cos(x))−1 = 1/cos(x) = sec(x) or secant of x, the multiplicative inverse (or reciprocal) of the trigonometric function cosine (see above for ambiguity)
cos x−1, sometimes interpreted as cos(x−1) = cos(1/x), the cosine of the multiplicative inverse (or reciprocal) of x (see below for ambiguity)
cos x−1, sometimes interpreted as (cos(x))−1 = 1/cos(x) = sec(x) or secant of x, the multiplicative inverse (or reciprocal) of the trigonometric function cosine (see above for ambiguity)[citation needed]

Cos 1 Radian

If you want to know what is cos 1 radian in terms of trigonometry, then navigate straight to the explanations in the next paragraph; what’s ahead in this section is the value of cos 1:

cos1 = 0.5403cos 1 = 0.5403

cos 1 radian = 0.5403

The cos of 1 radian is 0.5403, the same as cos of 1 radian in degrees. To change 1 radian to degrees multiply 1 by by 180° / $\pi$ = 57.29578°. Cos 1 = cos 57.29578 degrees.

Our results of cos1 have been rounded to five decimal places. If you want cosine 1 with higher accuracy, then use the calculator below; our tool displays ten decimal places.

To calculate cos 1 radian insert the angle 1 in the field labelled rad, but if you want to calculate cos 1 in degrees, then you have to press the swap unit button first.A Really Cool Cosine Calculator and Useful Information! Please ReTweet. Click To TweetBesides cos1, similar trigonometric calculations on our site include, but are not limited, to:

Cos 86°
Cos -5.15
Cos -299°

The identities of cosine 1 are as follows:

cos1= sin (π/2 + 1) = sin 2.5707963267949

= sin (π/2 – 1) = sin 0.570796326794897

-cos1= cos (π + 1) = cos 4.14159265358979

= cos (π – 1) = cos 2.14159265358979Note that cos1 is periodic: cos (1 + n × 2π) = cos 1, n$\hspace{5px} \in \hspace{5px} \mathbb{Z}$.

The inverse cos of 1, ie cos-1(1) is a very special value for the inverse cosine function. Remember that cos -1(x) will give you the angle whose cosine is x

The Value of the Inverse Cos of 1

As you can see below, the inverse cos-1 (1) is 0° or, in radian measure, 0 . '1' represents the maximum value of the cosine function. It happens at 0 and then again at 2Π, 4Π, 6Π etc..

Explanation:

For cos 1 degrees, the angle 1° lies between 0° and 90° (First Quadrant). Since cosine function is positive in the first quadrant, thus cos 1° value = 0.9998476. . .
Since the cosine function is a periodic function, we can represent cos 1° as, cos 1 degrees = cos(1° + n × 360°), n ∈ Z. ⇒ cos 1° = cos 361° = cos 721°, and so on.

Note: Since, cosine is an even function, the value of cos(-1°) = cos(1°).

Methods to Find Value of Cos 1 Degrees

The cosine function is positive in the 1st quadrant. The value of cos 1° is given as 0.99984. . .. We can find the value of cos 1 degrees by:

Using Unit Circle
Using Trigonometric Functions

Cos 1 Degrees Using Unit Circle

To find the value of cos 1 degrees using the unit circle:

Rotate ‘r’ anticlockwise to form 1° angle with the positive x-axis.
The cos of 1 degrees equals the x-coordinate(0.9998) of the point of intersection (0.9998, 0.0175) of unit circle and r.

Hence the value of cos 1° = x = 0.9998 (approx)

Cos 1° in Terms of Trigonometric Functions

Using trigonometry formulas, we can represent the cos 1 degrees as:

± √(1-sin²(1°))
± 1/√(1 + tan²(1°))
± cot 1°/√(1 + cot²(1°))
±√(cosec²(1°) - 1)/cosec 1°
1/sec 1°

Note: Since 1° lies in the 1st Quadrant, the final value of cos 1° will be positive.

We can use trigonometric identities to represent cos 1° as,

-cos(180° - 1°) = -cos 179°
-cos(180° + 1°) = -cos 181°
sin(90° + 1°) = sin 91°
sin(90° - 1°) = sin 89°

☛ Also Check:

Cos 1 degrees is the value of cosine trigonometric function for an angle equal to 1 degrees. The value of cos 1° is 0.9998 (approx)

How to Find Cos 1° in Terms of Other Trigonometric Functions?

Using trigonometry formula, the value of cos 1° can be given in terms of other trigonometric functions as:

± √(1-sin²(1°))
± 1/√(1 + tan²(1°))
± cot 1°/√(1 + cot²(1°))
± √(cosec²(1°) - 1)/cosec 1°
1/sec 1°

☛ Also check: trigonometry table

How to Find the Value of Cos 1 Degrees?

The value of cos 1 degrees can be calculated by constructing an angle of 1° with the x-axis, and then finding the coordinates of the corresponding point (0.9998, 0.0175) on the unit circle. The value of cos 1° is equal to the x-coordinate (0.9998). ∴ cos 1° = 0.9998.