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Tan 690 Degrees
The value of tan 690 degrees is 0.5773502. . .. Tan 690 degrees in radians is written as tan (690° × π/180°), i.e., tan (23π/6) or tan (12.042771. . .). In this article, we will discuss the methods to find the value of tan 690 degrees with examples.
 Tan 690°: 1/√3
 Tan 690° in decimal: 0.5773502. . .
 Tan (690 degrees): 0.5773502. . . or 1/√3
 Tan 690° in radians: tan (23π/6) or tan (12.0427718 . . .)
What is the Value of Tan 690 Degrees?
The value of tan 690 degrees in decimal is 0.577350269. . .. Tan 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 . . .
∴ tan 690° = tan(12.0427) = 1/√3 or 0.5773502. . .
Explanation:
For tan 690°, the angle 690° > 360°. We can represent tan 690° as, tan(690° mod 360°) = tan(330°). The angle 690°, coterminal to angle 330°, is located in the Fourth Quadrant(Quadrant IV).
Since tangent function is negative in the 4th quadrant, thus tan 690 degrees value = 1/√3 or 0.5773502. . .
Similarly, given the periodic property of tan 690°, it can also be written as, tan 690 degrees = (690° + n × 180°), n ∈ Z.
⇒ tan 690° = tan 870° = tan 1050°, and so on.
Note: Since, tangent is an odd function, the value of tan(690°) = tan(690°).
Methods to Find Value of Tan 690 Degrees
The tangent function is negative in the 4th quadrant. The value of tan 690° is given as 0.57735. . .. We can find the value of tan 690 degrees by:
 Using Trigonometric Functions
 Using Unit Circle
Tan 690° in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the tan 690 degrees as:
 sin(690°)/cos(690°)
 ± sin 690°/√(1  sin²(690°))
 ± √(1  cos²(690°))/cos 690°
 ± 1/√(cosec²(690°)  1)
 ± √(sec²(690°)  1)
 1/cot 690°
Note: Since 690° lies in the 4th Quadrant, the final value of tan 690° will be negative.
We can use trigonometric identities to represent tan 690° as,
 cot(90°  690°) = cot(600°)
 cot(90° + 690°) = cot 780°
 tan (180°  690°) = tan(510°)
Tan 690 Degrees Using Unit Circle
To find the value of tan 690 degrees using the unit circle, represent 690° in the form (1 × 360°) + 330° [∵ 690°>360°] ∵ The angle 690° is coterminal to 330° angle and also tangent is a periodic function, tan 690° = tan 330°.
 Rotate ‘r’ anticlockwise to form 330° or 690° angle with the positive xaxis.
 The tan of 690 degrees equals the ycoordinate(0.5) divided by xcoordinate(0.866) of the point of intersection (0.866, 0.5) of unit circle and r.
Hence the value of tan 690° = y/x = 0.5774 (approx).
☛ Also Check:
Examples Using Tan 690 Degrees

Example 1: Using the value of tan 690°, solve: (sec²(690°)  1).
Solution:
We know, (sec²(690°)  1) = (tan²(690°)) = 0.3333
⇒ (sec²(690°)  1) = 0.3333 
Example 2: Find the value of 5 tan(690°)/6 tan(510°).
Solution:
Using trigonometric identities, we know, tan(690°) = tan(180°  690°) = tan(510°).
⇒ tan(690°) = tan(510°)
⇒ Value of 5 tan(690°)/6 tan(510°) = 5/6 
Example 3: Simplify: 8 (tan 690°/cot(600°))
Solution:
We know tan 690° = cot(600°)
⇒ 8 tan 690°/cot(600°) = 8 (tan 690°/tan 690°)
= 8(1) = 8
FAQs on Tan 690 Degrees
What is Tan 690 Degrees?
Tan 690 degrees is the value of tangent trigonometric function for an angle equal to 690 degrees. The value of tan 690° is 1/√3 or 0.5774 (approx).
How to Find Tan 690° in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of tan 690° can be given in terms of other trigonometric functions as:
 sin(690°)/cos(690°)
 ± sin 690°/√(1  sin²(690°))
 ± √(1  cos²(690°))/cos 690°
 ± 1/√(cosec²(690°)  1)
 ± √(sec²(690°)  1)
 1/cot 690°
☛ Also check: trigonometric table
How to Find the Value of Tan 690 Degrees?
The value of tan 690 degrees can be calculated by constructing an angle of 690° with the xaxis, and then finding the coordinates of the corresponding point (0.866, 0.5) on the unit circle. The value of tan 690° is equal to the ycoordinate(0.5) divided by the xcoordinate (0.866). ∴ tan 690° = 1/√3 or 0.5774
What is the Value of Tan 690° in Terms of Cosec 690°?
Since the tangent function can be represented using the cosecant function, we can write tan 690° as 1/√(cosec²(690°)  1). The value of cosec 690° is equal to 2.
What is the Value of Tan 690 Degrees in Terms of Cot 690°?
Since the tangent function is the reciprocal of the cotangent function, we can write tan 690° as 1/cot(690°). The value of cot 690° is equal to √3.
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