How is sin related to cos

Product identities. Aside: weirdly enough, these product identities were used before logarithms were invented in order to perform multiplication. Average those two cosines. You get the product xy! Three table look-ups, and computing a sum, a difference, and an average rather than one multiplication.

Tycho Brahe — , among others, used this algorithm known as prosthaphaeresis. Triple angle formulas. You can easily reconstruct these from the addition and double angle formulas. More half-angle formulas. In a right triangle, the sine of one acute angle, A , equals the cosine of the other acute angle, B.

Yes, there is a "relationship" regarding the tangent of the two acute angles A and B in a right triangle. It is not, however, the same type of relationship that exists between sine and cosine. The angles being used in the following examples are acute angles.

Definition 1 is the simplest and most intuitive definition of the sine and cosine function. Sine and Cosine Topics overview formal definition graph properties expansions derivative integral. This second equation says that if we evaluate the cosine of that angle q , we will get the exact same value as if we divided the length of the side adjacent to that angle by the length of the triangle's hypotenuse.

These are the cofunction identities :. The definitions of sine and cosine can be rearranged a little bit to let you write down the lengths of the sides in terms of the hypotenuse and the angles. Can you write another expression for length b , one that uses a sine instead of a cosine?

Can you see how to write down two expressions for the length of side a? What is the length of side b? Hint: draw a picture, and label A , c , and b.

Solution: Pictures are always good. Always remember the rule that the side with a given letter is opposite the angle with that letter. And, conventionally, we always put C at the right angle, so that makes c the hypotenuse. Once you have the picture, solving the problem is pretty straightforward. You want something involving A , its adjacent side, and the hypotenuse; that has to be the cosine.

Example: A guy wire is anchored in the ground and attached to the top of a foot flagpole. Which function involves the opposite side and the hypotenuse? It must be the sine. You may be wondering how to find sides or angles of triangles when there is no right angle.

One important special case comes up frequently. In other words, in a unit right triangle the opposite side will equal the sine and the adjacent side will equal the cosine of the angle. The triangle is often drawn in a unit circle , a circle of radius 1, as shown at right.