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 x-axis.
• The tan of 5pi/4 equals the y-coordinate(-0.7071) divided by the x-coordinate(-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:

• sin 11pi/2
• tan 7pi/6
• sin pi/4
• cot pi/2
• cos 7pi/6
• cos pi/2

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 x-axis, 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 y-coordinate(-0.7071) divided by the x-coordinate (-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