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Tan 5pi/4
The value of tan 5pi/4 is 1. Tan 5pi/4 radians in degrees is written as tan ((5π/4) × 180°/π), i.e., tan (225°). In this article, we will discuss the methods to find the value of tan 5pi/4 with examples.
 Tan 5pi/4: 1
 Tan (5pi/4): 1
 Tan 5pi/4 in degrees: tan (225°)
What is the Value of Tan 5pi/4?
The value of tan 5pi/4 is 1. Tan 5pi/4 can also be expressed using the equivalent of the given angle (5pi/4) in degrees (225°).
We know, using radian to degree conversion, θ in degrees = θ in radians × (180°/pi)
⇒ 5pi/4 radians = 5pi/4 × (180°/pi) = 225° or 225 degrees
∴ tan 5pi/4 = tan 5π/4 = tan(225°) = 1
Explanation:
For tan 5pi/4, the angle 5pi/4 lies between pi and 3pi/2 (Third Quadrant). Since tangent function is positive in the third quadrant, thus tan 5pi/4 value = 1
Since the tangent function is a periodic function, we can represent tan 5pi/4 as, tan 5pi/4 = tan(5pi/4 + n × pi), n ∈ Z.
⇒ tan 5pi/4 = tan 9pi/4 = tan 13pi/4 , and so on.
Note: Since, tangent is an odd function, the value of tan(5pi/4) = tan(5pi/4).
Methods to Find Value of Tan 5pi/4
The tangent function is positive in the 3rd quadrant. The value of tan 5pi/4 is given as 1. We can find the value of tan 5pi/4 by:
 Using Trigonometric Functions
 Using Unit Circle
Tan 5pi/4 in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the tan 5pi/4 as:
 sin(5pi/4)/cos(5pi/4)
 ± sin(5pi/4)/√(1  sin²(5pi/4))
 ± √(1  cos²(5pi/4))/cos(5pi/4)
 ± 1/√(cosec²(5pi/4)  1)
 ± √(sec²(5pi/4)  1)
 1/cot(5pi/4)
Note: Since 5pi/4 lies in the 3rd Quadrant, the final value of tan 5pi/4 will be positive.
We can use trigonometric identities to represent tan 5pi/4 as,
 cot(pi/2  5pi/4) = cot(3pi/4)
 cot(pi/2 + 5pi/4) = cot 7pi/4
 tan (pi  5pi/4) = tan(pi/4)
Tan 5pi/4 Using Unit Circle
To find the value of tan 5π/4 using the unit circle:
 Rotate ‘r’ anticlockwise to form 5pi/4 angle with the positive xaxis.
 The tan of 5pi/4 equals the ycoordinate(0.7071) divided by the xcoordinate(0.7071) of the point of intersection (0.7071, 0.7071) of unit circle and r.
Hence the value of tan 5pi/4 = y/x = 1
☛ Also Check:
Examples Using Tan 5pi/4

Example 1: Using the value of tan 5pi/4, solve: (sec²(5pi/4)  1).
Solution:
We know, (sec²(5pi/4)  1) = (tan²(5pi/4)) = 1
⇒ (sec²(5pi/4)  1) = 1 
Example 2: Find the value of (2 sin (5pi/8) cos (5pi/8) sec (5pi/4)). [Hint: Use tan 5pi/4 = 1]
Solution:
Using sin 2a formula,
2 sin (5pi/8) cos (5pi/8) = sin (2 × 5pi/8) = sin 5pi/4
⇒ 2 sin (5pi/8) cos (5pi/8) sec(5pi/4) = sin(5pi/4) sec(5pi/4)
= sin(5pi/4)/cos(5pi/4) = tan 5pi/4
⇒ (2 sin (5pi/8) cos (5pi/8) sec(5pi/4)) = 1 
Example 3: Simplify: 9 (tan(5pi/4)/cot(3pi/4))
Solution:
We know tan 5pi/4 = cot(3pi/4)
⇒ 9 tan(5pi/4)/cot(3pi/4) = 9 tan(5pi/4)/tan(5pi/4)
= 9(1) = 9
FAQs on Tan 5pi/4
What is Tan 5pi/4?
Tan 5pi/4 is the value of tangent trigonometric function for an angle equal to 5π/4 radians. The value of tan 5pi/4 is 1.
What is the Value of Tan 5pi/4 in Terms of Sin 5pi/4?
Using trigonometric identities, we can write tan 5pi/4 in terms of sin 5pi/4 as, tan(5pi/4) = sin(5pi/4)/√(1  sin²(5pi/4)) . Here, the value of sin 5pi/4 is equal to (1/√2).
How to Find the Value of Tan 5pi/4?
The value of tan 5pi/4 can be calculated by constructing an angle of 5π/4 radians with the xaxis, and then finding the coordinates of the corresponding point (0.7071, 0.7071) on the unit circle. The value of tan 5pi/4 is equal to the ycoordinate(0.7071) divided by the xcoordinate (0.7071). ∴ tan 5pi/4 = 1
What is the Value of Tan 5pi/4 in Terms of Cosec 5pi/4?
Since the tangent function can be represented using the cosecant function, we can write tan 5pi/4 as 1/√(cosec²(5pi/4)  1). The value of cosec 5pi/4 is equal to 1.41421.
How to Find Tan 5pi/4 in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of tan 5pi/4 can be given in terms of other trigonometric functions as:
 sin(5pi/4)/cos(5pi/4)
 ± sin(5pi/4)/√(1  sin²(5pi/4))
 ± √(1  cos²(5pi/4))/cos(5pi/4)
 ± 1/√(cosec²(5pi/4)  1)
 ± √(sec²(5pi/4)  1)
 1/cot(5pi/4)
☛ Also check: trigonometry table
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