Sin 54 Degrees

The value of sin 54 degrees is 0.8090169. . .. Sin 54 degrees in radians is written as sin (54° × π/180°), i.e., sin (3π/10) or sin (0.942477. . .). In this article, we will discuss the methods to find the value of sin 54 degrees with examples.

• Sin 54°: 0.8090169. . .
• Sin 54° in fraction: (√5 + 1)/4
• Sin (-54 degrees): -0.8090169. . .
• Sin 54° in radians: sin (3π/10) or sin (0.9424777 . . .)

What is the Value of Sin 54 Degrees?

The value of sin 54 degrees in decimal is 0.809016994. . .. Sin 54 degrees can also be expressed using the equivalent of the given angle (54 degrees) in radians (0.94247 . . .).

We know, using degree to radian conversion, θ in radians = θ in degrees × (pi/180°)
⇒ 54 degrees = 54° × (π/180°) rad = 3π/10 or 0.9424 . . .
∴ sin 54° = sin(0.9424) = (√5 + 1)/4 or 0.8090169. . .

Explanation:

For sin 54 degrees, the angle 54° lies between 0° and 90° (First Quadrant). Since sine function is positive in the first quadrant, thus sin 54° value = (√5 + 1)/4 or 0.8090169. . .
Since the sine function is a periodic function, we can represent sin 54° as, sin 54 degrees = sin(54° + n × 360°), n ∈ Z.
⇒ sin 54° = sin 414° = sin 774°, and so on.
Note: Since, sine is an odd function, the value of sin(-54°) = -sin(54°).

Methods to Find Value of Sin 54 Degrees

The sine function is positive in the 1st quadrant. The value of sin 54° is given as 0.80901. . .. We can find the value of sin 54 degrees by:

• Using Unit Circle
• Using Trigonometric Functions

Sin 54 Degrees Using Unit Circle

To find the value of sin 54 degrees using the unit circle:

• Rotate ‘r’ anticlockwise to form a 54° angle with the positive x-axis.
• The sin of 54 degrees equals the y-coordinate(0.809) of the point of intersection (0.5878, 0.809) of unit circle and r.

Hence the value of sin 54° = y = 0.809 (approx)

Sin 54° in Terms of Trigonometric Functions

Using trigonometry formulas, we can represent the sin 54 degrees as:

• ± √(1-cos²(54°))
• ± tan 54°/√(1 + tan²(54°))
• ± 1/√(1 + cot²(54°))
• ± √(sec²(54°) - 1)/sec 54°
• 1/cosec 54°

Note: Since 54° lies in the 1st Quadrant, the final value of sin 54° will be positive.

We can use trigonometric identities to represent sin 54° as,

• sin(180° - 54°) = sin 126°
• -sin(180° + 54°) = -sin 234°
• cos(90° - 54°) = cos 36°
• -cos(90° + 54°) = -cos 144°

☛ Also Check:

• sin 11 degrees
• sin 35 degrees
• sin 18 degrees
• sin 300 degrees
• sin 4 degrees
• sin 72 degrees

FAQs on Sin 54 Degrees

What is Sin 54 Degrees?

Sin 54 degrees is the value of sine trigonometric function for an angle equal to 54 degrees. The value of sin 54° is (√5 + 1)/4 or 0.809 (approx).

What is the Value of Sin 54 Degrees in Terms of Cot 54°?

We can represent the sine function in terms of the cotangent function using trig identities, sin 54° can be written as 1/√(1 + cot²(54°)). Here, the value of cot 54° is equal to 0.72654.

How to Find Sin 54° in Terms of Other Trigonometric Functions?

Using trigonometry formula, the value of sin 54° can be given in terms of other trigonometric functions as:

• ± √(1-cos²(54°))
• ± tan 54°/√(1 + tan²(54°))
• ± 1/√(1 + cot²(54°))
• ± √(sec²(54°) - 1)/sec 54°
• 1/cosec 54°

☛ Also check: trigonometry table

How to Find the Value of Sin 54 Degrees?

The value of sin 54 degrees can be calculated by constructing an angle of 54° with the x-axis, and then finding the coordinates of the corresponding point (0.5878, 0.809) on the unit circle. The value of sin 54° is equal to the y-coordinate (0.809). ∴ sin 54° = 0.809.

What is the Exact Value of sin 54 Degrees?

The exact value of sin 54 degrees can be given accurately up to 8 decimal places as 0.80901699 and (√5 + 1)/4 in fraction.