What Are the Trigonometric Formulas?
Question
Trigonometric formulas relate the geometry of triangles to the functions of angles. Usually these functions are
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What Are the Trigonometric Formulas?

Hi, my name is Bassem Saad. I'm an associate math instructor and a Ph.D. candidate, and I'm here today for About.com to introduce the trigonometric formulas. Trigonometric formulas relate the geometry of triangles to the functions of angles. Usually these functions are sines, cosines, and tangents.
Right Triangle Trigonometric Formulas

The first set of formulas we're going to look at will relate only to the geometry of right triangles. So a right triangle is any triangle that has 90 degrees as one of its angles. So the first formula would relate the sine of theta (that's an angle that we'll specify on the triangle) to the edge opposite of theta, divided by the hypotenuse - that is the angle opposite of 90 degrees. The next one is the cosine of theta, which is the side adjacent to the angle theta, divided by the hypotenuse. And the last one is tangent of theta, which is the side opposite of the angle, divided by the side adjacent to the angle.

Let's take a look at an example: Given an angle of 45 degrees, find the tangent if the side opposite has unit length two and the side adjacent has unit length two. Well then, we look at our formula; that's just two, divided by two. That tells us that the tangent of 45 degrees equals one.
Cosine Trigonometric Formulas

Another important set of trigonometric formulas are the cosine formulas – that's these formulas; they work on any triangle. And the way we'll label our triangle are the lower-case alphabet letters: a, b, and c represents the angles of the triangle. And the upper-case, capital A, would represent the opposite edge of lower-case a, and we do that all the way around.

So let's take a look at an example. We'll let lower-case a be 60 degrees, capital B be two unit lengths, capital C be three unit lengths, and we want to know what the length of capital A is. So looking at these formulas, we actually only know the angle of lower-case a, so it should be the top formula to tell us what the length of capital A is. We can plug in all the values, and then we have to remember that the cosine of 60 degrees is one-half. After plugging in the values, we evaluate and we find that the length of A is four.
Sine Trigonometric Formulas

So let's take a look at our last set of important trigonometric formulas: that's the sine formulas, and again they work for any triangle, and we'll be using the same labeling system.

Let's take a look at an example. We'll have a equal 45 degrees, the angle of c equals 30 degrees, the length of capital A is one, the length of capital C is the square root of two. And let's say we knew that the sine of 30 degrees is one-half, but we didn't know the sine of 45 degrees. We can use the sine formula to find the answer; that is, we've got one, divided by the sine of 30 degrees, equals the square root of two, divided by the sine of 45 degrees. Using algebra, we solve for 45 degrees, and we evaluate to get the sine of 45 degrees equals the square root of two, divided by two.

So now we know what the trigonometric formulas are. Thanks for watching, and to learn more visit us on the web at About.com.