Sin 780 Degrees

The value of sin 780 degrees is 0.8660254. . .. Sin 780 degrees in radians is written as sin (780° × π/180°), i.e., sin (13π/3) or sin (13.613568. . .). In this article, we will discuss the methods to find the value of sin 780 degrees with examples.

• Sin 780°: 0.8660254. . .
• Sin 780° in fraction: √3/2
• Sin (-780 degrees): -0.8660254. . .
• Sin 780° in radians: sin (13π/3) or sin (13.6135681 . . .)

What is the Value of Sin 780 Degrees?

The value of sin 780 degrees in decimal is 0.866025403. . .. Sin 780 degrees can also be expressed using the equivalent of the given angle (780 degrees) in radians (13.61356 . . .).

We know, using degree to radian conversion, θ in radians = θ in degrees × (pi/180°)
⇒ 780 degrees = 780° × (π/180°) rad = 13π/3 or 13.6135 . . .
∴ sin 780° = sin(13.6135) = √3/2 or 0.8660254. . .

Explanation:

For sin 780°, the angle 780° > 360°. Given the periodic property of the sine function, we can represent it as sin(780° mod 360°) = sin(60°). The angle 780°, coterminal to angle 60°, is located in the First Quadrant(Quadrant I).
Since sine function is positive in the 1st quadrant, thus sin 780 degrees value = √3/2 or 0.8660254. . .
Similarly, sin 780° can also be written as, sin 780 degrees = (780° + n × 360°), n ∈ Z.
⇒ sin 780° = sin 1140° = sin 1500°, and so on.
Note: Since, sine is an odd function, the value of sin(-780°) = -sin(780°).

Methods to Find Value of Sin 780 Degrees

The sine function is positive in the 1st quadrant. The value of sin 780° is given as 0.86602. . .. We can find the value of sin 780 degrees by:

• Using Unit Circle
• Using Trigonometric Functions

Sin 780 Degrees Using Unit Circle

To find the value of sin 780 degrees using the unit circle, represent 780° in the form (2 × 360°) + 60° [∵ 780°>360°] ∵ sine is a periodic function, sin 780° = sin 60°.

• Rotate ‘r’ anticlockwise to form a 60° or 780° angle with the positive x-axis.
• The sin of 780 degrees equals the y-coordinate(0.866) of the point of intersection (0.5, 0.866) of unit circle and r.

Hence the value of sin 780° = y = 0.866 (approx)

Sin 780° in Terms of Trigonometric Functions

Using trigonometry formulas, we can represent the sin 780 degrees as:

• ± √(1-cos²(780°))
• ± tan 780°/√(1 + tan²(780°))
• ± 1/√(1 + cot²(780°))
• ± √(sec²(780°) - 1)/sec 780°
• 1/cosec 780°

Note: Since 780° lies in the 1st Quadrant, the final value of sin 780° will be positive.

We can use trigonometric identities to represent sin 780° as,

• sin(180° - 780°) = sin(-600°)
• -sin(180° + 780°) = -sin 960°
• cos(90° - 780°) = cos(-690°)
• -cos(90° + 780°) = -cos 870°

☛ Also Check:

• sin 36 degrees
• sin 30 degrees
• sin 104 degrees
• sin 960 degrees
• sin 16 degrees
• sin 836 degrees

FAQs on Sin 780 Degrees

What is Sin 780 Degrees?

Sin 780 degrees is the value of sine trigonometric function for an angle equal to 780 degrees. The value of sin 780° is √3/2 or 0.866 (approx).

How to Find the Value of Sin 780 Degrees?

The value of sin 780 degrees can be calculated by constructing an angle of 780° with the x-axis, and then finding the coordinates of the corresponding point (0.5, 0.866) on the unit circle. The value of sin 780° is equal to the y-coordinate (0.866). ∴ sin 780° = 0.866.

What is the Value of Sin 780 Degrees in Terms of Cos 780°?

Using trigonometric identities, we can write sin 780° in terms of cos 780° as, sin(780°) = √(1-cos²(780°)). Here, the value of cos 780° is equal to 0.5.

How to Find Sin 780° in Terms of Other Trigonometric Functions?

Using trigonometry formula, the value of sin 780° can be given in terms of other trigonometric functions as:

• ± √(1-cos²(780°))
• ± tan 780°/√(1 + tan²(780°))
• ± 1/√(1 + cot²(780°))
• ± √(sec²(780°) - 1)/sec 780°
• 1/cosec 780°

☛ Also check: trigonometry table

What is the Exact Value of sin 780 Degrees?

The exact value of sin 780 degrees can be given accurately up to 8 decimal places as 0.86602540 and √3/2 in fraction.