Sin 72 Degrees

The value of sin 72 degrees is 0.9510565. . .. Sin 72 degrees in radians is written as sin (72° × π/180°), i.e., sin (2π/5) or sin (1.256637. . .). In this article, we will discuss the methods to find the value of sin 72 degrees with examples.

• Sin 72°: 0.9510565. . .
• Sin 72° in fraction: √(10 - 2√5)/2
• Sin (-72 degrees): -0.9510565. . .
• Sin 72° in radians: sin (2π/5) or sin (1.2566370 . . .)

What is the Value of Sin 72 Degrees?

The value of sin 72 degrees in decimal is 0.951056516. . .. Sin 72 degrees can also be expressed using the equivalent of the given angle (72 degrees) in radians (1.25663 . . .).

We know, using degree to radian conversion, θ in radians = θ in degrees × (pi/180°)
⇒ 72 degrees = 72° × (π/180°) rad = 2π/5 or 1.2566 . . .
∴ sin 72° = sin(1.2566) = √(10 - 2√5)/2 or 0.9510565. . .

Explanation:

For sin 72 degrees, the angle 72° lies between 0° and 90° (First Quadrant). Since sine function is positive in the first quadrant, thus sin 72° value = √(10 - 2√5)/2 or 0.9510565. . .
Since the sine function is a periodic function, we can represent sin 72° as, sin 72 degrees = sin(72° + n × 360°), n ∈ Z.
⇒ sin 72° = sin 432° = sin 792°, and so on.
Note: Since, sine is an odd function, the value of sin(-72°) = -sin(72°).

Methods to Find Value of Sin 72 Degrees

The sine function is positive in the 1st quadrant. The value of sin 72° is given as 0.95105. . .. We can find the value of sin 72 degrees by:

• Using Trigonometric Functions
• Using Unit Circle

Sin 72° in Terms of Trigonometric Functions

Using trigonometry formulas, we can represent the sin 72 degrees as:

• ± √(1-cos²(72°))
• ± tan 72°/√(1 + tan²(72°))
• ± 1/√(1 + cot²(72°))
• ± √(sec²(72°) - 1)/sec 72°
• 1/cosec 72°

Note: Since 72° lies in the 1st Quadrant, the final value of sin 72° will be positive.

We can use trigonometric identities to represent sin 72° as,

• sin(180° - 72°) = sin 108°
• -sin(180° + 72°) = -sin 252°
• cos(90° - 72°) = cos 18°
• -cos(90° + 72°) = -cos 162°

Sin 72 Degrees Using Unit Circle

To find the value of sin 72 degrees using the unit circle:

• Rotate ‘r’ anticlockwise to form a 72° angle with the positive x-axis.
• The sin of 72 degrees equals the y-coordinate(0.9511) of the point of intersection (0.309, 0.9511) of unit circle and r.

Hence the value of sin 72° = y = 0.9511 (approx)

☛ Also Check:

• sin 30 degrees
• sin 36 degrees
• sin 50 degrees
• sin 9 degrees
• sin 720 degrees
• sin 144 degrees

FAQs on Sin 72 Degrees

What is Sin 72 Degrees?

Sin 72 degrees is the value of sine trigonometric function for an angle equal to 72 degrees. The value of sin 72° is √(10 - 2√5)/2 or 0.9511 (approx).

How to Find Sin 72° in Terms of Other Trigonometric Functions?

Using trigonometry formula, the value of sin 72° can be given in terms of other trigonometric functions as:

• ± √(1-cos²(72°))
• ± tan 72°/√(1 + tan²(72°))
• ± 1/√(1 + cot²(72°))
• ± √(sec²(72°) - 1)/sec 72°
• 1/cosec 72°

☛ Also check: trigonometry table

How to Find the Value of Sin 72 Degrees?

The value of sin 72 degrees can be calculated by constructing an angle of 72° with the x-axis, and then finding the coordinates of the corresponding point (0.309, 0.9511) on the unit circle. The value of sin 72° is equal to the y-coordinate (0.9511). ∴ sin 72° = 0.9511.

What is the Value of Sin 72 Degrees in Terms of Cot 72°?

We can represent the sine function in terms of the cotangent function using trig identities, sin 72° can be written as 1/√(1 + cot²(72°)). Here, the value of cot 72° is equal to 0.32491.

What is the Value of Sin 72° in Terms of Sec 72°?

Since the sine function can be represented using the secant function, we can write sin 72° as √(sec²(72°) - 1)/sec 72°. The value of sec 72° is equal to 3.236068.