Cos 165 Degrees

The value of cos 165 degrees is -0.9659258. . .. Cos 165 degrees in radians is written as cos (165° × π/180°), i.e., cos (11π/12) or cos (2.879793. . .). In this article, we will discuss the methods to find the value of cos 165 degrees with examples.

• Cos 165°: -0.9659258. . .
• Cos 165° in fraction: -(√6+√2)/4
• Cos (-165 degrees): -0.9659258. . .
• Cos 165° in radians: cos (11π/12) or cos (2.8797932 . . .)

What is the Value of Cos 165 Degrees?

The value of cos 165 degrees in decimal is -0.965925826. . .. Cos 165 degrees can also be expressed using the equivalent of the given angle (165 degrees) in radians (2.87979 . . .)

We know, using degree to radian conversion, θ in radians = θ in degrees × (pi/180°)
⇒ 165 degrees = 165° × (π/180°) rad = 11π/12 or 2.8797 . . .
∴ cos 165° = cos(2.8797) = -(√6+√2)/4 or -0.9659258. . .

Explanation:

For cos 165 degrees, the angle 165° lies between 90° and 180° (Second Quadrant). Since cosine function is negative in the second quadrant, thus cos 165° value = -(√6+√2)/4 or -0.9659258. . .
Since the cosine function is a periodic function, we can represent cos 165° as, cos 165 degrees = cos(165° + n × 360°), n ∈ Z.
⇒ cos 165° = cos 525° = cos 885°, and so on.
Note: Since, cosine is an even function, the value of cos(-165°) = cos(165°).

Methods to Find Value of Cos 165 Degrees

The cosine function is negative in the 2nd quadrant. The value of cos 165° is given as -0.96592. . .. We can find the value of cos 165 degrees by:

• Using Unit Circle
• Using Trigonometric Functions

Cos 165 Degrees Using Unit Circle

To find the value of cos 165 degrees using the unit circle:

• Rotate ‘r’ anticlockwise to form 165° angle with the positive x-axis.
• The cos of 165 degrees equals the x-coordinate(-0.9659) of the point of intersection (-0.9659, 0.2588) of unit circle and r.

Hence the value of cos 165° = x = -0.9659 (approx)

Cos 165° in Terms of Trigonometric Functions

Using trigonometry formulas, we can represent the cos 165 degrees as:

• ± √(1-sin²(165°))
• ± 1/√(1 + tan²(165°))
• ± cot 165°/√(1 + cot²(165°))
• ±√(cosec²(165°) - 1)/cosec 165°
• 1/sec 165°

Note: Since 165° lies in the 2nd Quadrant, the final value of cos 165° will be negative.

We can use trigonometric identities to represent cos 165° as,

• -cos(180° - 165°) = -cos 15°
• -cos(180° + 165°) = -cos 345°
• sin(90° + 165°) = sin 255°
• sin(90° - 165°) = sin(-75°)

☛ Also Check:

• cos 150 degrees
• cos 11 degrees
• cos 0 degrees
• cos 73 degrees
• cos 1170 degrees
• cos 13 degrees

FAQs on Cos 165 Degrees

What is Cos 165 Degrees?

Cos 165 degrees is the value of cosine trigonometric function for an angle equal to 165 degrees. The value of cos 165° is -(√6+√2)/4 or -0.9659 (approx)

How to Find Cos 165° in Terms of Other Trigonometric Functions?

Using trigonometry formula, the value of cos 165° can be given in terms of other trigonometric functions as:

• ± √(1-sin²(165°))
• ± 1/√(1 + tan²(165°))
• ± cot 165°/√(1 + cot²(165°))
• ± √(cosec²(165°) - 1)/cosec 165°
• 1/sec 165°

☛ Also check: trigonometric table

What is the Value of Cos 165 Degrees in Terms of Cot 165°?

We can represent the cosine function in terms of the cotangent function using trig identities, cos 165° can be written as cot 165°/√(1 + cot²(165°)). Here, the value of cot 165° is equal to -3.73205.

How to Find the Value of Cos 165 Degrees?

The value of cos 165 degrees can be calculated by constructing an angle of 165° with the x-axis, and then finding the coordinates of the corresponding point (-0.9659, 0.2588) on the unit circle. The value of cos 165° is equal to the x-coordinate (-0.9659). ∴ cos 165° = -0.9659.

What is the Value of Cos 165° in Terms of Sec 165°?

Since the secant function is the reciprocal of the cosine function, we can write cos 165° as 1/sec(165°). The value of sec 165° is equal to -1.035276.