Complementary relationship:
The complementary relationship between sinθ and cosθ is as follows: sinθ = 1/cosθ, cosθ = 1/sinθ.
In other words, the sine and cosine values of the two angles are reciprocal.
Complementary relationship:
The complementary relationship between sinθ and cos (90-θ) is as follows: sinθ = cos(90 -θ) and cosθ = sin(90 -θ).
In other words, the sine of one angle is equal to the cosine of the other angle.
These relations are often used in the calculation and application of trigonometric functions, and they can be used interchangeably, which is convenient for solving problems and simplifying.
The complementary relationship between sin and cos is widely used in mathematics and physics.
Transformation between 1. trigonometric functions: The complementary relationship can be used to transform sin and cos. By taking the reciprocal, the value of one trigonometric function can be expressed as the reciprocal form of another trigonometric function, thus simplifying calculation or solving problems.
2. Proof and derivation of trigonometric identities: Complementary relations can be used to prove and deduce various trigonometric identities. Using one trigonometric function to represent another trigonometric function can simplify the expression and get the equivalent form.
3. Complementarity of angles: Complementarity is used to describe the relationship between two angles, which is very useful in solving geometric problems and trigonometric functions. For example, when two straight lines intersect, their included angle and complementary angle satisfy a complementary relationship.
4. Analysis of physical fluctuation and vibration: In physics, sine and cosine functions are used to describe the phenomenon of fluctuation and vibration. Complementary relationship is often used in the analysis of fluctuation and vibration, and can be easily converted and compared between different angles, frequencies and amplitudes.
Examples of complementary relationship between sin and cos
Question: Given the sine value of angle A as 1/3, find the cosine value of angle A and the sine value of complementary angle B. ..
Solution:
According to the complementary relationship between sin and cos, sinθ = 1/cosθ. We know that sin A = 1/3, so we can get cos A = 3/ 1 = 3.
According to the complementary relationship between sin and cos, sinθ = cos(90 -θ), we require the sin value of complementary angle B of angle A, that is, SIN B. According to the complementary relationship, there is sin A = cos B, so sin B = sin A = 1/3.
Therefore, the cos value of angle A is 3, and the sin value of complementary angle B of angle A is 1/3.
This example shows the application of the complementary relationship between sin and cos in solving angle and trigonometric function values. By making full use of the complementary relationship, we can easily deduce other related trigonometric functions from a known trigonometric function value.