How to Determine the Angle θ When tan θ 4/3

The problem of finding the angle θ when tanθtan{theta} 4/3 is a fundamental task in trigonometry. This article guides you through the process step-by-step, detailing how to use the arctangent function and providing a comprehensive explanation with examples.

Using the Arctangent Function

To find the angle θ, you can use the arctan43arctanleft(frac{4}{3}right) function, which is also known as the inverse tangent function. This function finds the angle whose tangent is a given value.

Calculating the Angle

Here’s a step-by-step guide:

Use the arctangent function to find the angle in degrees or radians: θ {text{arctan}}left(frac{4}{3}right)theta {text{arctan}}left(frac{4}{3}right) Using a calculator, ensure it’s set to the appropriate mode (degrees or radians): If in degrees: θ ≈ 53.13°theta approx 53.13^circ If in radians: θ ≈ 0.93 text{ radians}theta approx 0.93 text{ radians}

Considering the Quadrant

The tangent function is positive in both the first and third quadrants. Therefore, the general solutions for θ are:

First Quadrant: θ ≈ 53.13°theta approx 53.13^circ or 0.930.93 radians Third Quadrant: θ ≈ 53.13° 180° 233.13°theta approx 53.13^circ 180^circ 233.13^circ or 0.93 pi approx 4.070.93 pi approx 4.07 radians

Using Trigonometric Tables

Much like the arctangent function, trigonometric tables can also be used to find the angle. Here’s how:

Convert the decimal to a fraction: 43 1.3333 (to 4 decimal places)frac{4}{3} 1.3333 ; (text{to 4 decimal places}) Locate 1.3333 in a table of natural tangents. You might find the closest value is 1.3335. This corresponds to an angle of 53°8′. Alternatively, use a scientific calculator: Input 43frac{4}{3} Press the Inv tan or arctan button The result will be the angle θ in degrees or radians

In summary, to determine the angle θ when tanθtan{theta} 4/3, you can use the arctangent function or trigonometric tables. Ensure your calculator is set to the correct mode and use the general solutions for both the first and third quadrants.