Cos 105 Degrees
The value of cos 105 degrees is -0.2588190. . .. Cos 105 degrees in radians is written as cos (105° × π/180°), i.e., cos (7π/12) or cos (1.832595. . .). In this article, we will discuss the methods to find the value of cos 105 degrees with examples.
- Cos 105°: -0.2588190. . .
- Cos 105° in fraction: -(√6-√2)/4
- Cos (-105 degrees): -0.2588190. . .
- Cos 105° in radians: cos (7π/12) or cos (1.8325957 . . .)
What is the Value of Cos 105 Degrees?
The value of cos 105 degrees in decimal is -0.258819045. . .. Cos 105 degrees can also be expressed using the equivalent of the given angle (105 degrees) in radians (1.83259 . . .)
We know, using degree to radian conversion, θ in radians = θ in degrees × (pi/180°)
⇒ 105 degrees = 105° × (π/180°) rad = 7π/12 or 1.8325 . . .
∴ cos 105° = cos(1.8325) = -(√6-√2)/4 or -0.2588190. . .
Explanation:
For cos 105 degrees, the angle 105° lies between 90° and 180° (Second Quadrant). Since cosine function is negative in the second quadrant, thus cos 105° value = -(√6-√2)/4 or -0.2588190. . .
Since the cosine function is a periodic function, we can represent cos 105° as, cos 105 degrees = cos(105° + n × 360°), n ∈ Z.
⇒ cos 105° = cos 465° = cos 825°, and so on.
Note: Since, cosine is an even function, the value of cos(-105°) = cos(105°).
Methods to Find Value of Cos 105 Degrees
The cosine function is negative in the 2nd quadrant. The value of cos 105° is given as -0.25881. . .. We can find the value of cos 105 degrees by:
- Using Unit Circle
- Using Trigonometric Functions
Cos 105 Degrees Using Unit Circle
To find the value of cos 105 degrees using the unit circle:
- Rotate ‘r’ anticlockwise to form 105° angle with the positive x-axis.
- The cos of 105 degrees equals the x-coordinate(-0.2588) of the point of intersection (-0.2588, 0.9659) of unit circle and r.
Hence the value of cos 105° = x = -0.2588 (approx)
Cos 105° in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the cos 105 degrees as:
- ± √(1-sin²(105°))
- ± 1/√(1 + tan²(105°))
- ± cot 105°/√(1 + cot²(105°))
- ±√(cosec²(105°) - 1)/cosec 105°
- 1/sec 105°
Note: Since 105° lies in the 2nd Quadrant, the final value of cos 105° will be negative.
We can use trigonometric identities to represent cos 105° as,
- -cos(180° - 105°) = -cos 75°
- -cos(180° + 105°) = -cos 285°
- sin(90° + 105°) = sin 195°
- sin(90° - 105°) = sin(-15°)
☛ Also Check:
FAQs on Cos 105 Degrees
What is Cos 105 Degrees?
Cos 105 degrees is the value of cosine trigonometric function for an angle equal to 105 degrees. The value of cos 105° is -(√6-√2)/4 or -0.2588 (approx)
What is the Value of Cos 105 Degrees in Terms of Sin 105°?
Using trigonometric identities, we can write cos 105° in terms of sin 105° as, cos(105°) = -√(1 - sin²(105°)). Here, the value of sin 105° is equal to (√6 + √2)/4.
How to Find Cos 105° in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of cos 105° can be given in terms of other trigonometric functions as:
- ± √(1-sin²(105°))
- ± 1/√(1 + tan²(105°))
- ± cot 105°/√(1 + cot²(105°))
- ± √(cosec²(105°) - 1)/cosec 105°
- 1/sec 105°
☛ Also check: trigonometric table
What is the Value of Cos 105° in Terms of Cosec 105°?
Since the cosine function can be represented using the cosecant function, we can write cos 105° as -[√(cosec²(105°) - 1)/cosec 105°]. The value of cosec 105° is equal to 1.03527.
How to Find the Value of Cos 105 Degrees?
The value of cos 105 degrees can be calculated by constructing an angle of 105° with the x-axis, and then finding the coordinates of the corresponding point (-0.2588, 0.9659) on the unit circle. The value of cos 105° is equal to the x-coordinate (-0.2588). ∴ cos 105° = -0.2588.