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Sin 165 Degrees

The value of sin 165 degrees is 0.2588190. . .. Sin 165 degrees in radians is written as sin (165° × π/180°), i.e., sin (11π/12) or sin (2.879793. . .). In this article, we will discuss the methods to find the value of sin 165 degrees with examples.

Sin 165°: 0.2588190. . .
Sin 165° in fraction: (√6 - √2)/4
Sin (-165 degrees): -0.2588190. . .
Sin 165° in radians: sin (11π/12) or sin (2.8797932 . . .)

What is the Value of Sin 165 Degrees?

The value of sin 165 degrees in decimal is 0.258819045. . .. Sin 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 . . .
∴ sin 165° = sin(2.8797) = (√6 - √2)/4 or 0.2588190. . .

Explanation:

For sin 165 degrees, the angle 165° lies between 90° and 180° (Second Quadrant). Since sine function is positive in the second quadrant, thus sin 165° value = (√6 - √2)/4 or 0.2588190. . .
Since the sine function is a periodic function, we can represent sin 165° as, sin 165 degrees = sin(165° + n × 360°), n ∈ Z.
⇒ sin 165° = sin 525° = sin 885°, and so on.
Note: Since, sine is an odd function, the value of sin(-165°) = -sin(165°).

Methods to Find Value of Sin 165 Degrees

The sine function is positive in the 2nd quadrant. The value of sin 165° is given as 0.25881. . .. We can find the value of sin 165 degrees by:

Using Trigonometric Functions
Using Unit Circle

Sin 165° in Terms of Trigonometric Functions

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

± √(1-cos²(165°))
± tan 165°/√(1 + tan²(165°))
± 1/√(1 + cot²(165°))
± √(sec²(165°) - 1)/sec 165°
1/cosec 165°

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

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

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

Sin 165 Degrees Using Unit Circle

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

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

Hence the value of sin 165° = y = 0.2588 (approx)

☛ Also Check:

Examples Using Sin 165 Degrees

Example 1: Using the value of sin 165°, solve: (1-cos²(165°)).

Solution:
We know, (1-cos²(165°)) = (sin²(165°)) = 0.067
⇒ (1-cos²(165°)) = 0.067
Example 2: Find the value of 5 sin(165°)/7 cos(-75°).

Solution:

Using trigonometric identities, we know, sin(165°) = cos(90° - 165°) = cos(-75°).
⇒ sin(165°) = cos(-75°)
⇒ Value of 5 sin(165°)/7 cos(-75°) = 5/7
Example 3: Simplify: 2 (sin 165°/sin 525°)

Solution:

We know sin 165° = sin 525°
⇒ 2 sin 165°/sin 525° = 2(sin 165°/sin 165°)
= 2(1) = 2

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FAQs on Sin 165 Degrees

What is Sin 165 Degrees?

Sin 165 degrees is the value of sine trigonometric function for an angle equal to 165 degrees. The value of sin 165° is (√6 - √2)/4 or 0.2588 (approx).

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

Since the sine function can be represented using the secant function, we can write sin 165° as -√(sec²(165°) - 1)/sec 165°. The value of sec 165° is equal to -1.035276.

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

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

How to Find the Value of Sin 165 Degrees?

The value of sin 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 sin 165° is equal to the y-coordinate (0.2588). ∴ sin 165° = 0.2588.

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

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

± √(1-cos²(165°))
± tan 165°/√(1 + tan²(165°))
± 1/√(1 + cot²(165°))
± √(sec²(165°) - 1)/sec 165°
1/cosec 165°

☛ Also check: trigonometry table

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