Tan 405 Degrees
The value of tan 405 degrees is 1. Tan 405 degrees in radians is written as tan (405° × π/180°), i.e., tan (9π/4) or tan (7.068583. . .). In this article, we will discuss the methods to find the value of tan 405 degrees with examples.
- Tan 405°: 1
- Tan (-405 degrees): -1
- Tan 405° in radians: tan (9π/4) or tan (7.0685834 . . .)
What is the Value of Tan 405 Degrees?
The value of tan 405 degrees is 1. Tan 405 degrees can also be expressed using the equivalent of the given angle (405 degrees) in radians (7.06858 . . .)
We know, using degree to radian conversion, θ in radians = θ in degrees × (pi/180°)
⇒ 405 degrees = 405° × (π/180°) rad = 9π/4 or 7.0685 . . .
∴ tan 405° = tan(7.0685) = 1
Explanation:
For tan 405°, the angle 405° > 360°. We can represent tan 405° as, tan(405° mod 360°) = tan(45°). The angle 405°, coterminal to angle 45°, is located in the First Quadrant(Quadrant I).
Since tangent function is positive in the 1st quadrant, thus tan 405 degrees value = 1
Similarly, given the periodic property of tan 405°, it can also be written as, tan 405 degrees = (405° + n × 180°), n ∈ Z.
⇒ tan 405° = tan 585° = tan 765°, and so on.
Note: Since, tangent is an odd function, the value of tan(-405°) = -tan(405°).
Methods to Find Value of Tan 405 Degrees
The tangent function is positive in the 1st quadrant. The value of tan 405° is given as 1. We can find the value of tan 405 degrees by:
- Using Unit Circle
- Using Trigonometric Functions
Tan 405 Degrees Using Unit Circle
To find the value of tan 405 degrees using the unit circle, represent 405° in the form (1 × 360°) + 45° [∵ 405°>360°] ∵ The angle 405° is coterminal to 45° angle and also tangent is a periodic function, tan 405° = tan 45°.
- Rotate ‘r’ anticlockwise to form 45° or 405° angle with the positive x-axis.
- The tan of 405 degrees equals the y-coordinate(0.7071) divided by x-coordinate(0.7071) of the point of intersection (0.7071, 0.7071) of unit circle and r.
Hence the value of tan 405° = y/x = 1
Tan 405° in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the tan 405 degrees as:
- sin(405°)/cos(405°)
- ± sin 405°/√(1 - sin²(405°))
- ± √(1 - cos²(405°))/cos 405°
- ± 1/√(cosec²(405°) - 1)
- ± √(sec²(405°) - 1)
- 1/cot 405°
Note: Since 405° lies in the 1st Quadrant, the final value of tan 405° will be positive.
We can use trigonometric identities to represent tan 405° as,
- cot(90° - 405°) = cot(-315°)
- -cot(90° + 405°) = -cot 495°
- -tan (180° - 405°) = -tan(-225°)
☛ Also Check:
FAQs on Tan 405 Degrees
What is Tan 405 Degrees?
Tan 405 degrees is the value of tangent trigonometric function for an angle equal to 405 degrees. The value of tan 405° is 1.
How to Find Tan 405° in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of tan 405° can be given in terms of other trigonometric functions as:
- sin(405°)/cos(405°)
- ± sin 405°/√(1 - sin²(405°))
- ± √(1 - cos²(405°))/cos 405°
- ± 1/√(cosec²(405°) - 1)
- ± √(sec²(405°) - 1)
- 1/cot 405°
☛ Also check: trigonometry table
What is the Value of Tan 405 Degrees in Terms of Sin 405°?
Using trigonometric identities, we can write tan 405° in terms of sin 405° as, tan(405°) = sin 405°/√(1 - sin²(405°)) . Here, the value of sin 405° is equal to 0.7071.
How to Find the Value of Tan 405 Degrees?
The value of tan 405 degrees can be calculated by constructing an angle of 405° with the x-axis, and then finding the coordinates of the corresponding point (0.7071, 0.7071) on the unit circle. The value of tan 405° is equal to the y-coordinate(0.7071) divided by the x-coordinate (0.7071). ∴ tan 405° = 1
What is the Value of Tan 405° in Terms of Sec 405°?
We can represent the tangent function in terms of the secant function using trig identities, tan 405° can be written as √(sec²(405°) - 1). Here, the value of sec 405° is equal to 1.4142.