Understanding the Inverse Tangent of 4/3 in Degrees and Radians

The inverse tangent, or arctangent, of a ratio is a common function in trigonometry. This piece explains how to find the value of tan^{-1}(4/3) both in degrees and radians. Understanding this function is crucial for solving various mathematical and real-world problems, such as in physics, engineering, and computer graphics.

Calculating tan^{-1}(4/3) in Degrees

To find the value of tan^{-1}(4/3) in degrees, we can use a calculator or trigonometric tables:

Note: Using a calculator, you find that:

tan^{-1}(4/3) ≈ 53.13°

This value can be expressed in approximate degrees as:

tan^{-1}(4/3) ≈ 53.13°

Converting Degrees to Radians

For many applications, especially in advanced mathematics and computing, angles are often expressed in radians rather than degrees. We can convert the approximate degree value to radians using the conversion factor:

53.13° × (π/180) ≈ 0.927 radians

Hence, the value of tan^{-1}(4/3) is approximately 53.13° or 0.927 radians.

Location in the Unit Circle

Given that the tangent function in this case is a positive value, the corresponding angle in the unit circle is in the first quadrant, not the third quadrant as mentioned in the original text. The Pythagorean triple 3-4-5 is a well-known right triangle, where:

x 3 y 4 hypotenuse 5

The tangent of this angle is the ratio of the opposite side to the adjacent side:

tan θ  4/3

This angle, however, is not π/4 (which is 57.295779513°), but it is related to it. The exact angle is approximately 53.13°.

Inverse Tangent of 4/3 in Radians

The inverse tangent of 4/3 in radians is approximately 0.927 radians. This value arises from the angle in the third quadrant where the tangent is also positive, given that tan(π θ) tan(θ).

Specifically, the angle 4π/3 is in the third quadrant, where both sine and cosine are negative, but the tangent is positive:

sin(4π/3) -√3/2 cos(4π/3) -1/2 tan(4π/3) √3

Thus, the value of the angle is:

tan^{-1}(4/3)  4π/3

This angle is not one of the commonly memorized angles, such as π/4 or π/6, but it is important to understand how to find it using a calculator or trigonometric tables.

Conclusion

In summary, tan^{-1}(4/3) in degrees is approximately 53.13°, and in radians, it is approximately 0.927 radians. Understanding these concepts is crucial for working with trigonometric functions, and it is essential to distinguish between measuring angles in degrees and radians for accurate calculations.