Understanding the Behavior of sin(tan(cos(x))) in Degrees

In this article, we delve into the intriguing behavior of the expression sin(tan(cos(x))) when x is measured in degrees. We explore the mechanics and the reasons behind the surprising result that this expression tends to approach approximately 0.0174 for small angles, but varies widely as x grows.

The Components of the Expression

To properly understand why sin(tan(cos(x))) converges to 0.0174, it is crucial to break down the individual functions: tangent, cosine, and sine.

Tangent Function: The tangent function, (tan(x)), can take any angle x in degrees and provide a value that can vary widely. However, for angles close to 0 degrees, (tan(x)) is approximately equal to x in radians, which is a small value.

Cosine Function

The cosine function, (cos(y)), for a small angle in radians is approximately equal to 1 minus the square of that angle divided by 2. Thus, for small values of y, (cos(y) approx 1 - frac{y^2}{2}). This approximation allows us to see that if (tan(x)) is small, (cos(tan(x))) will be close to 1.

Sine Function

The sine function, (sin(z)), for an angle close to 1 radian can be approximated as (sin(z) approx z) for small z. Since the cosine function outputs values between 0 and 1, (sin(cos(tan(x)))) will also yield a small output.

Approaching 0.0174

The number 0.0174 is significant because it is approximately equal to (sin(1^circ)). Let's explore how this value is connected to the expression in question:

For small angles, (tan(x)) can yield values that, when passed through the cosine function, approach values near (1^circ) or its radian equivalent, which is approximately 0.01745. Thus, for certain small angles, (cos(tan(x))) can be close to (1^circ) in radians, leading to (sin(cos(tan(x)))) being close to (sin(1^circ)), which is approximately (0.0174).

Conclusion

So, when evaluating sin(tan(cos(x))) for small angles x, the output tends to approach 0.0174 as x approaches 0 degrees due to the behavior of the tangent, cosine, and sine functions in that range. However, as x increases beyond small angles, the output will vary more widely based on the specific angle chosen.

It is important to note that the expression does not always yield the same value. For example:

sincos89.5° ≈ -0.00726 For the function sincosx°, the output is always between -0.0174524 and 0.0174524. For all real α: (-1 ≤ cos(α) ≤ 1) implies that (-1 ≤ cos(tan(x)°) ≤ 1).

Therefore:

-0.0174524 ≤ sincosx° ≤ 0.0174524

However, when x is not close to (90° 180n), then sincosx° approaches 0.0174 for a range of values such as: - For -85° ≤ x ≤ 85°, (tan(-85°) ≤ tan(x)° ≤ tan(85°)) results in -11.43 ≤ tan(x)° ≤ 11.43. Since (cos(x)) is decreasing in quadrant I, we have (0.98 ≤ cos(tan(x)°) ≤ 1) leading to (0.01710 ≤ sin(cos(tan(x)°)) ≤ 0.01745).

For (-80° ≤ x ≤ 80°,) we get (0.01737 ≤ sin(cos(tan(x)°)) ≤ 0.01745).

When x is close to (90° 180n,) then sincosx° oscillates wildly between -0.01745 and 0.01745.