Cos 75 Degrees
The value of cos 75 degrees is 0.2588190. . .. Cos 75 degrees in radians is written as cos (75° × π/180°), i.e., cos (5π/12) or cos (1.308996. . .). In this article, we will discuss the methods to find the value of cos 75 degrees with examples.
 Cos 75°: 0.2588190. . .
 Cos 75° in fraction: (√6  √2)/4
 Cos (75 degrees): 0.2588190. . .
 Cos 75° in radians: cos (5π/12) or cos (1.3089969 . . .)
What is the Value of Cos 75 Degrees?
The value of cos 75 degrees in decimal is 0.258819045. . .. Cos 75 degrees can also be expressed using the equivalent of the given angle (75 degrees) in radians (1.30899 . . .)
We know, using degree to radian conversion, θ in radians = θ in degrees × (pi/180°)
⇒ 75 degrees = 75° × (π/180°) rad = 5π/12 or 1.3089 . . .
∴ cos 75° = cos(1.3089) = (√6  √2)/4 or 0.2588190. . .
Explanation:
For cos 75 degrees, the angle 75° lies between 0° and 90° (First Quadrant). Since cosine function is positive in the first quadrant, thus cos 75° value = (√6  √2)/4 or 0.2588190. . .
Since the cosine function is a periodic function, we can represent cos 75° as, cos 75 degrees = cos(75° + n × 360°), n ∈ Z.
⇒ cos 75° = cos 435° = cos 795°, and so on.
Note: Since, cosine is an even function, the value of cos(75°) = cos(75°).
Methods to Find Value of Cos 75 Degrees
The cosine function is positive in the 1st quadrant. The value of cos 75° is given as 0.25881. . .. We can find the value of cos 75 degrees by:
 Using Trigonometric Functions
 Using Unit Circle
Cos 75° in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the cos 75 degrees as:
 ± √(1sin²(75°))
 ± 1/√(1 + tan²(75°))
 ± cot 75°/√(1 + cot²(75°))
 ±√(cosec²(75°)  1)/cosec 75°
 1/sec 75°
Note: Since 75° lies in the 1st Quadrant, the final value of cos 75° will be positive.
We can use trigonometric identities to represent cos 75° as,
 cos(180°  75°) = cos 105°
 cos(180° + 75°) = cos 255°
 sin(90° + 75°) = sin 165°
 sin(90°  75°) = sin 15°
Cos 75 Degrees Using Unit Circle
To find the value of cos 75 degrees using the unit circle:
 Rotate ‘r’ anticlockwise to form 75° angle with the positive xaxis.
 The cos of 75 degrees equals the xcoordinate(0.2588) of the point of intersection (0.2588, 0.9659) of unit circle and r.
Hence the value of cos 75° = x = 0.2588 (approx)
☛ Also Check:
Examples Using Cos 75 Degrees

Example 1: Find the value of (cos² 37.5°  sin² 37.5°). [Hint: Use cos 75° = 0.2588]
Solution:
Using the cos 2a formula,
(cos² 37.5°  sin² 37.5°) = cos(2 × 37.5°) = cos 75°
∵ cos 75° = 0.2588
⇒ (cos² 37.5°  sin² 37.5°) = 0.2588 
Example 2: Find the value of cos 75° if sec 75° is 3.8637.
Solution:
Since, cos 75° = 1/sec 75°
⇒ cos 75° = 1/3.8637 = 0.2588 
Example 3: Find the value of 2 cos(75°)/3 sin(15°).
Solution:
Using trigonometric identities, we know, cos(75°) = sin(90°  75°) = sin 15°.
⇒ cos(75°) = sin(15°)
⇒ Value of 2 cos(75°)/3 sin(15°) = 2/3
FAQs on Cos 75 Degrees
What is Cos 75 Degrees?
Cos 75 degrees is the value of cosine trigonometric function for an angle equal to 75 degrees. The value of cos 75° is (√6  √2)/4 or 0.2588 (approx)
How to Find Cos 75° in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of cos 75° can be given in terms of other trigonometric functions as:
 ± √(1sin²(75°))
 ± 1/√(1 + tan²(75°))
 ± cot 75°/√(1 + cot²(75°))
 ± √(cosec²(75°)  1)/cosec 75°
 1/sec 75°
☛ Also check: trigonometric table
How to Find the Value of Cos 75 Degrees?
The value of cos 75 degrees can be calculated by constructing an angle of 75° with the xaxis, and then finding the coordinates of the corresponding point (0.2588, 0.9659) on the unit circle. The value of cos 75° is equal to the xcoordinate (0.2588). ∴ cos 75° = 0.2588.
What is the Exact Value of cos 75 Degrees?
The exact value of cos 75 degrees can be given accurately up to 8 decimal places as 0.25881904 and (√6  √2)/4 in fraction.
What is the Value of Cos 75 Degrees in Terms of Cot 75°?
We can represent the cosine function in terms of the cotangent function using trig identities, cos 75° can be written as cot 75°/√(1 + cot²(75°)). Here, the value of cot 75° is equal to 0.26794.
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