Sin 690 Degrees
The value of sin 690 degrees is -0.5. Sin 690 degrees in radians is written as sin (690° × π/180°), i.e., sin (23π/6) or sin (12.042771. . .). In this article, we will discuss the methods to find the value of sin 690 degrees with examples.
- Sin 690°: -0.5
- Sin 690° in fraction: -(1/2)
- Sin (-690 degrees): 0.5
- Sin 690° in radians: sin (23π/6) or sin (12.0427718 . . .)
What is the Value of Sin 690 Degrees?
The value of sin 690 degrees in decimal is -0.5. Sin 690 degrees can also be expressed using the equivalent of the given angle (690 degrees) in radians (12.04277 . . .).
We know, using degree to radian conversion, θ in radians = θ in degrees × (pi/180°)
⇒ 690 degrees = 690° × (π/180°) rad = 23π/6 or 12.0427 . . .
∴ sin 690° = sin(12.0427) = -(1/2) or -0.5
Explanation:
For sin 690°, the angle 690° > 360°. Given the periodic property of the sine function, we can represent it as sin(690° mod 360°) = sin(330°). The angle 690°, coterminal to angle 330°, is located in the Fourth Quadrant(Quadrant IV).
Since sine function is negative in the 4th quadrant, thus sin 690 degrees value = -(1/2) or -0.5
Similarly, sin 690° can also be written as, sin 690 degrees = (690° + n × 360°), n ∈ Z.
⇒ sin 690° = sin 1050° = sin 1410°, and so on.
Note: Since, sine is an odd function, the value of sin(-690°) = -sin(690°).
Methods to Find Value of Sin 690 Degrees
The sine function is negative in the 4th quadrant. The value of sin 690° is given as -0.5. We can find the value of sin 690 degrees by:
- Using Trigonometric Functions
- Using Unit Circle
Sin 690° in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the sin 690 degrees as:
- ± √(1-cos²(690°))
- ± tan 690°/√(1 + tan²(690°))
- ± 1/√(1 + cot²(690°))
- ± √(sec²(690°) - 1)/sec 690°
- 1/cosec 690°
Note: Since 690° lies in the 4th Quadrant, the final value of sin 690° will be negative.
We can use trigonometric identities to represent sin 690° as,
- sin(180° - 690°) = sin(-510°)
- -sin(180° + 690°) = -sin 870°
- cos(90° - 690°) = cos(-600°)
- -cos(90° + 690°) = -cos 780°
Sin 690 Degrees Using Unit Circle
To find the value of sin 690 degrees using the unit circle, represent 690° in the form (1 × 360°) + 330° [∵ 690°>360°] ∵ sine is a periodic function, sin 690° = sin 330°.
- Rotate ‘r’ anticlockwise to form a 330° or 690° angle with the positive x-axis.
- The sin of 690 degrees equals the y-coordinate(-0.5) of the point of intersection (0.866, -0.5) of unit circle and r.
Hence the value of sin 690° = y = -0.5
☛ Also Check:
FAQs on Sin 690 Degrees
What is Sin 690 Degrees?
Sin 690 degrees is the value of sine trigonometric function for an angle equal to 690 degrees. The value of sin 690° is -(1/2) or -0.5.
How to Find Sin 690° in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of sin 690° can be given in terms of other trigonometric functions as:
- ± √(1-cos²(690°))
- ± tan 690°/√(1 + tan²(690°))
- ± 1/√(1 + cot²(690°))
- ± √(sec²(690°) - 1)/sec 690°
- 1/cosec 690°
☛ Also check: trigonometric table
What is the Value of Sin 690° in Terms of Sec 690°?
Since the sine function can be represented using the secant function, we can write sin 690° as -√(sec²(690°) - 1)/sec 690°. The value of sec 690° is equal to 1.154701.
How to Find the Value of Sin 690 Degrees?
The value of sin 690 degrees can be calculated by constructing an angle of 690° with the x-axis, and then finding the coordinates of the corresponding point (0.866, -0.5) on the unit circle. The value of sin 690° is equal to the y-coordinate (-0.5). ∴ sin 690° = -0.5.
What is the Value of Sin 690 Degrees in Terms of Cot 690°?
We can represent the sine function in terms of the cotangent function using trig identities, sin 690° can be written as -1/√(1 + cot²(690°)). Here, the value of cot 690° is equal to -1.73205.