Tan 75 Degrees
The value of tan 75 degrees is 3.7320508. . .. Tan 75 degrees in radians is written as tan (75° × π/180°), i.e., tan (5π/12) or tan (1.308996. . .). In this article, we will discuss the methods to find the value of tan 75 degrees with examples.
- Tan 75°: 2 + √3
- Tan 75° in decimal: 3.7320508. . .
- Tan (-75 degrees): -3.7320508. . . or -2 - √3
- Tan 75° in radians: tan (5π/12) or tan (1.3089969 . . .)
What is the Value of Tan 75 Degrees?
The value of tan 75 degrees in decimal is 3.732050807. . .. Tan 75 degrees can also be expressed using the equivalent of the given angle (75 degrees) in radians (1.30899 . . .)
We know, using degree to radian conversion, θ in radians = θ in degrees × (pi/180°)
⇒ 75 degrees = 75° × (π/180°) rad = 5π/12 or 1.3089 . . .
∴ tan 75° = tan(1.3089) = 2 + √3 or 3.7320508. . .
Explanation:
For tan 75 degrees, the angle 75° lies between 0° and 90° (First Quadrant). Since tangent function is positive in the first quadrant, thus tan 75° value = 2 + √3 or 3.7320508. . .
Since the tangent function is a periodic function, we can represent tan 75° as, tan 75 degrees = tan(75° + n × 180°), n ∈ Z.
⇒ tan 75° = tan 255° = tan 435°, and so on.
Note: Since, tangent is an odd function, the value of tan(-75°) = -tan(75°).
Methods to Find Value of Tan 75 Degrees
The tangent function is positive in the 1st quadrant. The value of tan 75° is given as 3.73205. . .. We can find the value of tan 75 degrees by:
- Using Trigonometric Functions
- Using Unit Circle
Tan 75° in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the tan 75 degrees as:
- sin(75°)/cos(75°)
- ± sin 75°/√(1 - sin²(75°))
- ± √(1 - cos²(75°))/cos 75°
- ± 1/√(cosec²(75°) - 1)
- ± √(sec²(75°) - 1)
- 1/cot 75°
Note: Since 75° lies in the 1st Quadrant, the final value of tan 75° will be positive.
We can use trigonometric identities to represent tan 75° as,
- cot(90° - 75°) = cot 15°
- -cot(90° + 75°) = -cot 165°
- -tan (180° - 75°) = -tan 105°
Tan 75 Degrees Using Unit Circle
To find the value of tan 75 degrees using the unit circle:
- Rotate ‘r’ anticlockwise to form 75° angle with the positive x-axis.
- The tan of 75 degrees equals the y-coordinate(0.9659) divided by x-coordinate(0.2588) of the point of intersection (0.2588, 0.9659) of unit circle and r.
Hence the value of tan 75° = y/x = 3.7321 (approx).
☛ Also Check:
FAQs on Tan 75 Degrees
What is Tan 75 Degrees?
Tan 75 degrees is the value of tangent trigonometric function for an angle equal to 75 degrees. The value of tan 75° is 2 + √3 or 3.7321 (approx).
How to Find Tan 75° in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of tan 75° can be given in terms of other trigonometric functions as:
- sin(75°)/cos(75°)
- ± sin 75°/√(1 - sin²(75°))
- ± √(1 - cos²(75°))/cos 75°
- ± 1/√(cosec²(75°) - 1)
- ± √(sec²(75°) - 1)
- 1/cot 75°
☛ Also check: trigonometry table
What is the Value of Tan 75° in Terms of Sec 75°?
We can represent the tangent function in terms of the secant function using trig identities, tan 75° can be written as √(sec²(75°) - 1). Here, the value of sec 75° is equal to 3.8637.
What is the Value of Tan 75 Degrees in Terms of Cot 75°?
Since the tangent function is the reciprocal of the cotangent function, we can write tan 75° as 1/cot(75°). The value of cot 75° is equal to 2 - √3.
How to Find the Value of Tan 75 Degrees?
The value of tan 75 degrees can be calculated by constructing an angle of 75° with the x-axis, and then finding the coordinates of the corresponding point (0.2588, 0.9659) on the unit circle. The value of tan 75° is equal to the y-coordinate(0.9659) divided by the x-coordinate (0.2588). ∴ tan 75° = 2 + √3 or 3.7321