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Tan 40 Degrees

The value of tan 40 degrees is 0.8390996. . .. Tan 40 degrees in radians is written as tan (40° × π/180°), i.e., tan (2π/9) or tan (0.698131. . .). In this article, we will discuss the methods to find the value of tan 40 degrees with examples.

Tan 40° in decimal: 0.8390996. . .
Tan (-40 degrees): -0.8390996. . .
Tan 40° in radians: tan (2π/9) or tan (0.6981317 . . .)

What is the Value of Tan 40 Degrees?

The value of tan 40 degrees in decimal is 0.839099631. . .. Tan 40 degrees can also be expressed using the equivalent of the given angle (40 degrees) in radians (0.69813 . . .)

We know, using degree to radian conversion, θ in radians = θ in degrees × (pi/180°)
⇒ 40 degrees = 40° × (π/180°) rad = 2π/9 or 0.6981 . . .
∴ tan 40° = tan(0.6981) = 0.8390996. . .

Explanation:

For tan 40 degrees, the angle 40° lies between 0° and 90° (First Quadrant). Since tangent function is positive in the first quadrant, thus tan 40° value = 0.8390996. . .
Since the tangent function is a periodic function, we can represent tan 40° as, tan 40 degrees = tan(40° + n × 180°), n ∈ Z.
⇒ tan 40° = tan 220° = tan 400°, and so on.
Note: Since, tangent is an odd function, the value of tan(-40°) = -tan(40°).

Methods to Find Value of Tan 40 Degrees

The tangent function is positive in the 1st quadrant. The value of tan 40° is given as 0.83909. . .. We can find the value of tan 40 degrees by:

Using Trigonometric Functions
Using Unit Circle

Tan 40° in Terms of Trigonometric Functions

Using trigonometry formulas, we can represent the tan 40 degrees as:

sin(40°)/cos(40°)
± sin 40°/√(1 - sin²(40°))
± √(1 - cos²(40°))/cos 40°
± 1/√(cosec²(40°) - 1)
± √(sec²(40°) - 1)
1/cot 40°

Note: Since 40° lies in the 1st Quadrant, the final value of tan 40° will be positive.

We can use trigonometric identities to represent tan 40° as,

cot(90° - 40°) = cot 50°
-cot(90° + 40°) = -cot 130°
-tan (180° - 40°) = -tan 140°

Tan 40 Degrees Using Unit Circle

To find the value of tan 40 degrees using the unit circle:

Rotate ‘r’ anticlockwise to form 40° angle with the positive x-axis.
The tan of 40 degrees equals the y-coordinate(0.6428) divided by x-coordinate(0.766) of the point of intersection (0.766, 0.6428) of unit circle and r.

Hence the value of tan 40° = y/x = 0.8391 (approx).

☛ Also Check:

Examples Using Tan 40 Degrees

Example 1: Find the value of (2 sin (20°) cos (20°) sec (40°)). [Hint: Use tan 40° = 0.8391]

Solution:

Using sin 2a formula,
2 sin (20°) cos (20°) = sin (2 × 20°) = sin 40°
⇒ 2 sin (20°) cos (20°) sec(40°) = sin 40° sec 40°
= sin 40°/cos 40° = tan 40°
⇒ (2 sin (20°) cos (20°) sec(40°)) = 0.8391
Example 2: Simplify: 5 (tan 40°/cot 50°)

Solution:

We know tan 40° = cot 50°
⇒ 5 tan 40°/cot 50° = 5 (tan 40°/tan 40°)
= 5(1) = 5
Example 3: Using the value of tan 40°, solve: (sec²(40°) - 1).

Solution:
We know, (sec²(40°) - 1) = (tan²(40°)) = 0.7041
⇒ (sec²(40°) - 1) = 0.7041

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FAQs on Tan 40 Degrees

What is Tan 40 Degrees?

Tan 40 degrees is the value of tangent trigonometric function for an angle equal to 40 degrees. The value of tan 40° is 0.8391 (approx).

How to Find the Value of Tan 40 Degrees?

The value of tan 40 degrees can be calculated by constructing an angle of 40° with the x-axis, and then finding the coordinates of the corresponding point (0.766, 0.6428) on the unit circle. The value of tan 40° is equal to the y-coordinate(0.6428) divided by the x-coordinate (0.766). ∴ tan 40° = 0.8391

What is the Value of Tan 40° in Terms of Cosec 40°?

Since the tangent function can be represented using the cosecant function, we can write tan 40° as 1/√(cosec²(40°) - 1). The value of cosec 40° is equal to 1.55572.

How to Find Tan 40° in Terms of Other Trigonometric Functions?

Using trigonometry formula, the value of tan 40° can be given in terms of other trigonometric functions as:

sin(40°)/cos(40°)
± sin 40°/√(1 - sin²(40°))
± √(1 - cos²(40°))/cos 40°
± 1/√(cosec²(40°) - 1)
± √(sec²(40°) - 1)
1/cot 40°

☛ Also check: trigonometric table

What is the Value of Tan 40 Degrees in Terms of Cos 40°?

We know, using trig identities, we can write tan 40° as √(1 - cos²(40°))/cos 40°. Here, the value of cos 40° is equal to 0.766044.

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