Tan 3pi/4

The value of tan 3pi/4 is -1. Tan 3pi/4 radians in degrees is written as tan ((3π/4) × 180°/π), i.e., tan (135°). In this article, we will discuss the methods to find the value of tan 3pi/4 with examples.

• Tan 3pi/4: -1
• Tan (-3pi/4): 1
• Tan 3pi/4 in degrees: tan (135°)

What is the Value of Tan 3pi/4?

The value of tan 3pi/4 is -1. Tan 3pi/4 can also be expressed using the equivalent of the given angle (3pi/4) in degrees (135°).

We know, using radian to degree conversion, θ in degrees = θ in radians × (180°/pi)
⇒ 3pi/4 radians = 3pi/4 × (180°/pi) = 135° or 135 degrees
∴ tan 3pi/4 = tan 3π/4 = tan(135°) = -1

Explanation:

For tan 3pi/4, the angle 3pi/4 lies between pi/2 and pi (Second Quadrant). Since tangent function is negative in the second quadrant, thus tan 3pi/4 value = -1
Since the tangent function is a periodic function, we can represent tan 3pi/4 as, tan 3pi/4 = tan(3pi/4 + n × pi), n ∈ Z.
⇒ tan 3pi/4 = tan 7pi/4 = tan 11pi/4 , and so on.
Note: Since, tangent is an odd function, the value of tan(-3pi/4) = -tan(3pi/4).

Methods to Find Value of Tan 3pi/4

The tangent function is negative in the 2nd quadrant. The value of tan 3pi/4 is given as -1. We can find the value of tan 3pi/4 by:

• Using Unit Circle
• Using Trigonometric Functions

Tan 3pi/4 Using Unit Circle

To find the value of tan 3π/4 using the unit circle:

• Rotate ‘r’ anticlockwise to form 3pi/4 angle with the positive x-axis.
• The tan of 3pi/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 3pi/4 = y/x = -1

Tan 3pi/4 in Terms of Trigonometric Functions

Using trigonometry formulas, we can represent the tan 3pi/4 as:

• sin(3pi/4)/cos(3pi/4)
• ± sin(3pi/4)/√(1 - sin²(3pi/4))
• ± √(1 - cos²(3pi/4))/cos(3pi/4)
• ± 1/√(cosec²(3pi/4) - 1)
• ± √(sec²(3pi/4) - 1)
• 1/cot(3pi/4)

Note: Since 3pi/4 lies in the 2nd Quadrant, the final value of tan 3pi/4 will be negative.

We can use trigonometric identities to represent tan 3pi/4 as,

• cot(pi/2 - 3pi/4) = cot(-pi/4)
• -cot(pi/2 + 3pi/4) = -cot 5pi/4
• -tan (pi - 3pi/4) = -tan pi/4

☛ Also Check:

• sin 17pi/12
• sin 4pi/3
• sec pi
• tan 11pi 6
• tan 19pi/3
• sin 5pi/12

FAQs on Tan 3pi/4

What is Tan 3pi/4?

Tan 3pi/4 is the value of tangent trigonometric function for an angle equal to 3π/4 radians. The value of tan 3pi/4 is -1.

How to Find Tan 3pi/4 in Terms of Other Trigonometric Functions?

Using trigonometry formula, the value of tan 3pi/4 can be given in terms of other trigonometric functions as:

• sin(3pi/4)/cos(3pi/4)
• ± sin(3pi/4)/√(1 - sin²(3pi/4))
• ± √(1 - cos²(3pi/4))/cos(3pi/4)
• ± 1/√(cosec²(3pi/4) - 1)
• ± √(sec²(3pi/4) - 1)
• 1/cot(3pi/4)

☛ Also check: trigonometric table

What is the Value of Tan 3pi/4 in Terms of Cosec 3pi/4?

Since the tangent function can be represented using the cosecant function, we can write tan 3pi/4 as -1/√(cosec²(3pi/4) - 1). The value of cosec 3pi/4 is equal to 1.41421.

What is the Value of Tan 3pi/4 in Terms of Sin 3pi/4?

Using trigonometric identities, we can write tan 3pi/4 in terms of sin 3pi/4 as, tan(3pi/4) = -sin(3pi/4)/√(1 - sin²(3pi/4)) . Here, the value of sin 3pi/4 is equal to 1/√2.

How to Find the Value of Tan 3pi/4?

The value of tan 3pi/4 can be calculated by constructing an angle of 3π/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 3pi/4 is equal to the y-coordinate(0.7071) divided by the x-coordinate (-0.7071). ∴ tan 3pi/4 = -1