Cos 5pi/6
The value of cos 5pi/6 is -0.8660254. . .. Cos 5pi/6 radians in degrees is written as cos ((5π/6) × 180°/π), i.e., cos (150°). In this article, we will discuss the methods to find the value of cos 5pi/6 with examples.
- Cos 5pi/6: −√3/2
- Cos 5pi/6 in decimal: -0.8660254. . .
- Cos (-5pi/6): -0.8660254. . . or -√(3)/2
- Cos 5pi/6 in degrees: cos (150°)
What is the Value of Cos 5pi/6?
The value of cos 5pi/6 in decimal is -0.866025403. . .. Cos 5pi/6 can also be expressed using the equivalent of the given angle (5pi/6) in degrees (150°).
We know, using radian to degree conversion, θ in degrees = θ in radians × (180°/pi)
⇒ 5pi/6 radians = 5pi/6 × (180°/pi) = 150° or 150 degrees
∴ cos 5pi/6 = cos 5π/6 = cos(150°) = −√3/2 or -0.8660254. . .
Explanation:
For cos 5pi/6, the angle 5pi/6 lies between pi/2 and pi (Second Quadrant). Since cosine function is negative in the second quadrant, thus cos 5pi/6 value = −√3/2 or -0.8660254. . .
Since the cosine function is a periodic function, we can represent cos 5pi/6 as, cos 5pi/6 = cos(5pi/6 + n × 2pi), n ∈ Z.
⇒ cos 5pi/6 = cos 17pi/6 = cos 29pi/6 , and so on.
Note: Since, cosine is an even function, the value of cos(-5pi/6) = cos(5pi/6).
Methods to Find Value of Cos 5pi/6
The cosine function is negative in the 2nd quadrant. The value of cos 5pi/6 is given as -0.86602. . .. We can find the value of cos 5pi/6 by:
- Using Trigonometric Functions
- Using Unit Circle
Cos 5pi/6 in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the cos 5pi/6 as:
- ± √(1-sin²(5pi/6))
- ± 1/√(1 + tan²(5pi/6))
- ± cot(5pi/6)/√(1 + cot²(5pi/6))
- ±√(cosec²(5pi/6) - 1)/cosec(5pi/6)
- 1/sec(5pi/6)
Note: Since 5pi/6 lies in the 2nd Quadrant, the final value of cos 5pi/6 will be negative.
We can use trigonometric identities to represent cos 5pi/6 as,
- -cos(pi - 5pi/6) = -cos pi/6
- -cos(pi + 5pi/6) = -cos 11pi/6
- sin(pi/2 + 5pi/6) = sin 4pi/3
- sin(pi/2 - 5pi/6) = sin(-pi/3)
Cos 5pi/6 Using Unit Circle
To find the value of cos 5π/6 using the unit circle:
- Rotate ‘r’ anticlockwise to form 5pi/6 angle with the positive x-axis.
- The cos of 5pi/6 equals the x-coordinate(-0.866) of the point of intersection (-0.866, 0.5) of unit circle and r.
Hence the value of cos 5pi/6 = x = -0.866 (approx)
☛ Also Check:
FAQs on Cos 5pi/6
What is Cos 5pi/6?
Cos 5pi/6 is the value of cosine trigonometric function for an angle equal to 5π/6 radians. The value of cos 5pi/6 is −√3/2 or -0.866 (approx)
What is the Value of Cos 5pi/6 in Terms of Sec 5pi/6?
Since the secant function is the reciprocal of the cosine function, we can write cos 5pi/6 as 1/sec(5pi/6). The value of sec 5pi/6 is equal to -1.154700.
How to Find Cos 5pi/6 in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of cos 5pi/6 can be given in terms of other trigonometric functions as:
- ± √(1-sin²(5pi/6))
- ± 1/√(1 + tan²(5pi/6))
- ± cot(5pi/6)/√(1 + cot²(5pi/6))
- ±√(cosec²(5pi/6) - 1)/cosec(5pi/6)
- 1/sec(5pi/6)
☛ Also check: trigonometric table
What is the Value of Cos 5pi/6 in Terms of Cot 5pi/6?
We can represent the cosine function in terms of the cotangent function using trig identities, cos 5pi/6 can be written as cot(5pi/6)/√(1 + cot²(5pi/6)). Here, the value of cot 5pi/6 is equal to -1.73205.
How to Find the Value of Cos 5pi/6?
The value of cos 5pi/6 can be calculated by constructing an angle of 5π/6 radians with the x-axis, and then finding the coordinates of the corresponding point (-0.866, 0.5) on the unit circle. The value of cos 5pi/6 is equal to the x-coordinate (-0.866). ∴ cos 5pi/6 = -0.866.