Exploring the Relationship Between Sin and Cos: Evaluating Specific Angles in Trigonometry

In the realm of mathematics, trigonometry is a core subject with numerous applications in various fields, from engineering to physics. Understanding the relationship between the sine and cosine functions is crucial for evaluating specific angles. In this article, we will explore the relationship between sin x and cos x, focusing specifically on the value of sin x when cos x equals 1/√2, and the angles where sin x equals cos x.

Value of Sin x When Cos x is Equal to 1/√2

We start with the fundamental trigonometric identity:

sin x ± √(1 - cos2 x)

Given that cos x 1/√2, we can substitute this value into the identity to find the corresponding sin x.

sin x ± √(1 - (1/√2)2)

sin x ± √(1 - 1/2)

sin x ± √(1/2)

sin x ± 1/√2

From this, we see that the sine function can be either positive or negative, depending on the quadrant in which the angle resides.

Quadrant Analysis

Since cos x is positive, we can determine which quadrants this holds for:

First Quadrant: Both sine and cosine are positive: cos x 1/√2, sin x 1/√2, hence x π/4 radians or 45°

Considering the periodic nature of trigonometric functions, we can generalize:

x n · 2π π/4 in radians x n · 360° 45° in degrees Fourth Quadrant: Cosine is positive, while sine is negative: cos x 1/√2, sin x -1/√2, hence x 7π/4 radians or 315°

Again, accounting for the periodicity:

x n · 2π - π/4 in radians x n · 360° - 45° in degrees

Sine Equals Cosine

Next, we explore the scenario where sin x cos x. To find the angles where this occurs, we can use the identity:

sin x / cos x 1

Since sin x / cos x tan x, we have:

tan x 1

tan x 1 for the angle x π/4 radians (or 45°) and has a period of π radians, so in general:

x n · π π/4 in radians x n · 180° 45° in degrees

where n is an integer.

Application in Right Triangles

Consider the right triangle with sides 1, √2, and adjacent side to the angle x equals 1. This forms a 45° angle because it is an isosceles triangle, with the hypotenuse being √2. Thus, we have:

cos x 1/√2 sin x 1/√2 hence, sin 45° cos 45°

Conclusion

In summary, the value of sin x when cos x is equal to 1/√2 is ±1/√2, which occurs in the first and fourth quadrants. Additionally, the angles where sin x cos x are found at 45° and every 180° thereafter. These relationships are fundamental to understanding the behavior of sine and cosine functions and their applications in various mathematical and scientific contexts.