Understanding the Trigonometric Identity: Understanding sin 59° - cos 31°

This article aims to provide a detailed explanation of the value of sin 59° - cos 31°, utilizing trigonometric identities. We will demonstrate how the value of this expression can be determined through a series of steps, using the identity that relates sine and cosine of complementary angles. By the end of this article, you will understand how to simplify trigonometric expressions and apply core trigonometric principles.

Step-by-Step Solution

To find the value of sin 59° - cos 31°, we can utilize the trigonometric identity that states:

cos(90° - θ) sin θ

Given that 31° 90° - 59°, we can apply the identity:

cos 31° sin 59°

With this in mind, we can substitute cos 31° with sin 59° in the original expression:

sin 59° - cos 31° sin 59° - sin 59° 0

Therefore, the value of sin 59° - cos 31° is 0.

Alternative Solutions and Explanations

Various solutions to the same trigonometric problem have been proposed, which are equivalent and based on the same principles:

T sin 60° - cos 30° T sin 90° - 30° - cos 30° T cos 30° - cos 30° T 0

These solutions are all derived from the same trigonometric identities and principles, such as the complementary angle identity cos(90° - x) sin x. Each step simplifies the expression to 0, reinforcing the fact that the value of sin 59° - cos 31° is 0.

Much Ado About Sine and Cosine

Trigonometry revolves around the relationships between angles and sides of triangles. Sine and cosine are two fundamental trigonometric functions that are defined based on the angles of a right-angled triangle:

Sine (sin x): is the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. Cosine (cos x): is the ratio of the length of the adjacent side to the hypotenuse in a right-angled triangle.

Understanding these relationships is crucial for solving trigonometric expressions and equations. The complementary angle identity, such as cos(90° - x) sin x, is a key tool in simplifying such expressions.

Conclusion

In conclusion, the value of sin 59° - cos 31° is 0. This result can be derived by leveraging the trigonometric identity cos(90° - θ) sin θ to relate the values of sine and cosine for complementary angles.

Keywords

Trigonometric identity, sine, cosine, complementary angles