Exploring Why Sine, Cosine, and Tangent Are Not Equal Despite the Same Arguments

The relationship between trigonometric functions like sine, cosine, and tangent can often lead to confusion, especially when the arguments seem the same. Let’s explore why these functions, although related to the same angle, do not yield the same values. We will delve into the unit circle and the ratios that define these functions, as well as look into some examples to solidify our understanding.

Defining Sine, Cosine, and Tangent

In trigonometry, the sine, cosine, and tangent of an angle θ are defined with respect to a right triangle. However, the unit circle, which is a circle of radius 1, provides a more comprehensive view of these functions. On the unit circle, an angle θ is measured from the positive x-axis, and its terminal side intersects the circle at a point (x, y).

The sine of an angle (sin θ) is the y-coordinate of the point of intersection with the unit circle. The cosine of an angle (cos θ) is the x-coordinate. The tangent (tan θ) is defined as the ratio of sine to cosine, i.e., sin θ / cos θ. This definition naturally leads to the difference in values between these functions.

The Unit Circle and Angle Arguments

Let's illustrate this with the unit circle. As the angle θ changes, the coordinates (x, y) corresponding to the point of intersection with the circle change, leading to different values for sine, cosine, and tangent.

Example 1: 0°

At 0°, the angle θ is measured from the positive x-axis. The point of intersection on the unit circle is (1, 0).

sin 0° y-coordinate 0 cos 0° x-coordinate 1 tan 0° sin 0° / cos 0° 0 / 1 0

Example 2: 90°

At 90°, the angle θ is measured from the positive x-axis, and the point of intersection on the unit circle is (0, 1).

sin 90° y-coordinate 1 cos 90° x-coordinate 0 tan 90° sin 90° / cos 90° 1 / 0 undefined (approaches infinity)

Understanding the Ratios

The ratios that define these functions also help us understand why they are not equal. Consider the sine and cosine functions:

sin θ opposite / hypotenuse cos θ adjacent / hypotenuse

The tangent is defined as the ratio of sine to cosine:

tan θ sin θ / cos θ

These ratios are the key to understanding why sine, cosine, and tangent have different values, even though the angle θ is the same. Let’s look at an example using a 45° angle:

Example 3: 45°

At 45°, the angle θ is measured from the positive x-axis, and the point of intersection on the unit circle is (√2/2, √2/2).

sin 45° y-coordinate √2/2 cos 45° x-coordinate √2/2 tan 45° sin 45° / cos 45° (√2/2) / (√2/2) 1

In this example, the sine and cosine values are identical, which is a special case. However, even in general, the different ratios mean they are not equal.

Conclusion

In conclusion, the values of sine, cosine, and tangent are not equal, even with the same angle argument, due to their definitions based on different ratios. This disparity is best understood through the unit circle and the specific definitions of these trigonometric functions. Understanding these concepts is crucial for anyone looking to master trigonometry and its applications in various fields, including physics, engineering, and more.

References:

Smith, D. E. (1951). iHistory of Mathematics, Vol. 2: Geometric Concepts to 1860/i. Dover Publications. Hipparchus, the father of trigonometry. Retrieved from