Tan pi/4
The value of tan pi/4 is 1. Tan pi/4 radians in degrees is written as tan ((π/4) × 180°/π), i.e., tan (45°). In this article, we will discuss the methods to find the value of tan pi/4 with examples.
- Tan pi/4: 1
- Tan (-pi/4): -1
- Tan pi/4 in degrees: tan (45°)
What is the Value of Tan pi/4?
The value of tan pi/4 is 1. Tan pi/4 can also be expressed using the equivalent of the given angle (pi/4) in degrees (45°).
We know, using radian to degree conversion, θ in degrees = θ in radians × (180°/pi)
⇒ pi/4 radians = pi/4 × (180°/pi) = 45° or 45 degrees
∴ tan pi/4 = tan π/4 = tan(45°) = 1
![Tan pi/4](https://static.qumath.in/static/website/old-cdn-static/trigonometry/tan-pi-4.png)
Explanation:
For tan pi/4, the angle pi/4 lies between 0 and pi/2 (First Quadrant). Since tangent function is positive in the first quadrant, thus tan pi/4 value = 1
Since the tangent function is a periodic function, we can represent tan pi/4 as, tan pi/4 = tan(pi/4 + n × pi), n ∈ Z.
⇒ tan pi/4 = tan 5pi/4 = tan 9pi/4 , and so on.
Note: Since, tangent is an odd function, the value of tan(-pi/4) = -tan(pi/4).
Methods to Find Value of Tan pi/4
The tangent function is positive in the 1st quadrant. The value of tan pi/4 is given as 1. We can find the value of tan pi/4 by:
- Using Trigonometric Functions
- Using Unit Circle
Tan pi/4 in Terms of Trigonometric Functions
Using trigonometry formulas, we can represent the tan pi/4 as:
- sin(pi/4)/cos(pi/4)
- ± sin(pi/4)/√(1 - sin²(pi/4))
- ± √(1 - cos²(pi/4))/cos(pi/4)
- ± 1/√(cosec²(pi/4) - 1)
- ± √(sec²(pi/4) - 1)
- 1/cot(pi/4)
Note: Since pi/4 lies in the 1st Quadrant, the final value of tan pi/4 will be positive.
We can use trigonometric identities to represent tan pi/4 as,
- cot(pi/2 - pi/4) = cot pi/4
- -cot(pi/2 + pi/4) = -cot 3pi/4
- -tan (pi - pi/4) = -tan 3pi/4
Tan pi/4 Using Unit Circle
![value of tan pi/4](https://static.qumath.in/static/website/old-cdn-static/trigonometry/value-of-tan-pi-4.png)
To find the value of tan π/4 using the unit circle:
- Rotate ‘r’ anticlockwise to form pi/4 angle with the positive x-axis.
- The tan of pi/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 pi/4 = y/x = 1
☛ Also Check:
Examples Using Tan pi/4
-
Example 1: Find the value of tan pi/4 if cot pi/4 is 1.
Solution:
Since, tan pi/4 = 1/cot(pi/4)
⇒ tan pi/4 = 1/1 = 1 -
Example 2: Find the value of (2 sin (pi/8) cos (pi/8) sec (pi/4)). [Hint: Use tan pi/4 = 1]
Solution:
Using sin 2a formula,
2 sin (pi/8) cos (pi/8) = sin (2 × pi/8) = sin pi/4
⇒ 2 sin (pi/8) cos (pi/8) sec(pi/4) = sin(pi/4) sec(pi/4)
= sin(pi/4)/cos(pi/4) = tan pi/4
⇒ (2 sin (pi/8) cos (pi/8) sec(pi/4)) = 1 -
Example 3: Simplify: 3 (tan(pi/4)/cot(pi/4))
Solution:
We know tan pi/4 = cot pi/4
⇒ 3 tan(pi/4)/cot(pi/4) = 3 tan(pi/4)/tan(pi/4)
= 3(1) = 3
FAQs on Tan pi/4
What is Tan pi/4?
Tan pi/4 is the value of tangent trigonometric function for an angle equal to π/4 radians. The value of tan pi/4 is 1.
How to Find Tan pi/4 in Terms of Other Trigonometric Functions?
Using trigonometry formula, the value of tan pi/4 can be given in terms of other trigonometric functions as:
- sin(pi/4)/cos(pi/4)
- ± sin(pi/4)/√(1 - sin²(pi/4))
- ± √(1 - cos²(pi/4))/cos(pi/4)
- ± 1/√(cosec²(pi/4) - 1)
- ± √(sec²(pi/4) - 1)
- 1/cot(pi/4)
☛ Also check: trigonometry table
What is the Value of Tan pi/4 in Terms of Cos pi/4?
We know, using trig identities, we can write tan pi/4 as √(1 - cos²(pi/4))/cos pi/4. Here, the value of cos pi/4 is equal to 0.707106.
How to Find the Value of Tan pi/4?
The value of tan pi/4 can be calculated by constructing an angle of π/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 pi/4 is equal to the y-coordinate(0.7071) divided by the x-coordinate (0.7071). ∴ tan pi/4 = 1
What is the Exact Value of tan pi/4?
The exact value of tan pi/4 is 1.
visual curriculum