from a handpicked tutor in LIVE 1to1 classes
Cos 45 Degrees
The value of cos 45 degrees is 0.7071067. . .. Cos 45 degrees in radians is written as cos (45° × π/180°), i.e., cos (π/4) or cos (0.785398. . .). In this article, we will discuss the methods to find the value of cos 45 degrees with examples.
 Cos 45°: 0.7071067. . .
 Cos 45° in fraction: 1/√2
 Cos (45 degrees): 0.7071067. . .
 Cos 45° in radians: cos (π/4) or cos (0.7853981 . . .)
What is the Value of Cos 45 Degrees?
The value of cos 45 degrees in decimal is 0.707106781. . .. Cos 45 degrees can also be expressed using the equivalent of the given angle (45 degrees) in radians (0.78539 . . .)
We know, using degree to radian conversion, θ in radians = θ in degrees × (pi/180°)
⇒ 45 degrees = 45° × (π/180°) rad = π/4 or 0.7853 . . .
∴ cos 45° = cos(0.7853) = 1/√2 or 0.7071067. . .
Explanation:
For cos 45 degrees, the angle 45° lies between 0° and 90° (First Quadrant). Since cosine function is positive in the first quadrant, thus cos 45° value = 1/√2 or 0.7071067. . .
Since the cosine function is a periodic function, we can represent cos 45° as, cos 45 degrees = cos(45° + n × 360°), n ∈ Z.
⇒ cos 45° = cos 405° = cos 765°, and so on.
Note: Since, cosine is an even function, the value of cos(45°) = cos(45°).
Methods to Find Value of Cos 45 Degrees
The cosine function is positive in the 1st quadrant. The value of cos 45° is given as 0.70710. . .. We can find the value of cos 45 degrees by:
 Using Unit Circle
 Using Trigonometric Functions
Cos 45 Degrees Using Unit Circle
To find the value of cos 45 degrees using the unit circle:
 Rotate ‘r’ anticlockwise to form 45° angle with the positive xaxis.
 The cos of 45 degrees equals the xcoordinate(0.7071) of the point of intersection (0.7071, 0.7071) of unit circle and r.
Hence the value of cos 45° = x = 0.7071 (approx)
Cos 45° in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the cos 45 degrees as:
 ± √(1sin²(45°))
 ± 1/√(1 + tan²(45°))
 ± cot 45°/√(1 + cot²(45°))
 ±√(cosec²(45°)  1)/cosec 45°
 1/sec 45°
Note: Since 45° lies in the 1st Quadrant, the final value of cos 45° will be positive.
We can use trigonometric identities to represent cos 45° as,
 cos(180°  45°) = cos 135°
 cos(180° + 45°) = cos 225°
 sin(90° + 45°) = sin 135°
 sin(90°  45°) = sin 45°
☛ Also Check:
Examples Using Cos 45 Degrees

Example 1: Using the value of cos 45°, solve: (1sin²(45°)).
Solution:
We know, (1sin²(45°)) = (cos²(45°)) = 0.5
⇒ (1sin²(45°)) = 0.5 
Example 2: Simplify: 5 (cos 45°/sin 135°)
Solution:
We know cos 45° = sin 135°
⇒ 5 cos 45°/sin 135° = 5 (cos 45°/cos 45°)
= 5(1) = 5 
Example 3: Find the value of (cos² 22.5°  sin² 22.5°). [Hint: Use cos 45° = 0.7071]
Solution:
Using the cos 2a formula,
(cos² 22.5°  sin² 22.5°) = cos(2 × 22.5°) = cos 45°
∵ cos 45° = 0.7071
⇒ (cos² 22.5°  sin² 22.5°) = 0.7071
FAQs on Cos 45 Degrees
What is Cos 45 Degrees?
Cos 45 degrees is the value of cosine trigonometric function for an angle equal to 45 degrees. The value of cos 45° is 1/√2 or 0.7071 (approx)
What is the Value of Cos 45 Degrees in Terms of Cot 45°?
We can represent the cosine function in terms of the cotangent function using trig identities, cos 45° can be written as cot 45°/√(1 + cot²(45°)). Here, the value of cot 45° is equal to 1.
How to Find Cos 45° in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of cos 45° can be given in terms of other trigonometric functions as:
 ± √(1sin²(45°))
 ± 1/√(1 + tan²(45°))
 ± cot 45°/√(1 + cot²(45°))
 ± √(cosec²(45°)  1)/cosec 45°
 1/sec 45°
☛ Also check: trigonometric table
What is the Value of Cos 45° in Terms of Cosec 45°?
Since the cosine function can be represented using the cosecant function, we can write cos 45° as [√(cosec²(45°)  1)/cosec 45°]. The value of cosec 45° is equal to 1.41421.
How to Find the Value of Cos 45 Degrees?
The value of cos 45 degrees can be calculated by constructing an angle of 45° with the xaxis, and then finding the coordinates of the corresponding point (0.7071, 0.7071) on the unit circle. The value of cos 45° is equal to the xcoordinate (0.7071). ∴ cos 45° = 0.7071.
visual curriculum