Final answer:
Knowledge of 45-degree angles provides the base sine and cosine values (√2/2), which can be used with trigonometric identities, double angle formulas, and the laws of sines and cosines to find the sine and cosine for other angles in a circle.
Step-by-step explanation:
To understand how the knowledge of 45-degree angles can help find the sine and cosine of other angles within a circle, we delve into the foundation of trigonometry.
For a 45-degree angle in a right triangle, both the sine and cosine values are √2/2 since the opposite and adjacent sides are equal in length.
However, knowing the sine and cosine for a 45-degree angle is just a starting point. When dealing with angles other than 45 degrees, you can use trigonometric identities and formulas to calculate the required values.
For example, the sine and cosine of a double angle, say 90 degrees, can be found using double angle formulas: sin(2·A) = 2sin(A)cos(A) and cos(2·A) = cos²(A) - sin²(A). By knowing the values of sine and cosine for 45 degrees, these could be applied directly to find the values for 90 degrees.
In the same vein, methods like the law of sines and the law of cosines are useful for calculating values in any type of triangle, extending beyond the right triangle scenario.
These laws relate the sides of a triangle to its angles and are crucial when the triangle in question is not a right triangle or when it is not practical to find a right angle to work from.
Additionally, understanding the sine and cosine functions on the unit circle can provide insight into the values of these functions for any angle, especially since the unit circle defines the sine and cosine in terms of x and y coordinates, which correlate to cosine and sine values respectively.