Trigonometry

Trigonometry is the branch of mathematics that studies triangles and their parts - angles and sides. Trigonometry is the study of triangles, their angles, and their sides. It is especially about right triangles, but you extend it to apply to all triangles.

Triangle

  • A triangle has three sides.
  • A triangle has three angles.
  • The three angles in a triangle add up to 180 degrees. So if you know two angles, you get the third angle by subtracting those two from 180 degrees.
  • With the Pythagorean theorem, you can find the length of one side of a right triangle when you know the length of the other two sides.

When you know three of a triangle's six parts (three sides and three angles), you can get all six parts, provided one of the given parts is the length of a side. This is called solving the triangle.

Trigonometric Functions

Six trigonometric functions describe the relationships between the sides of a triangle. Three are very important, and the other three are reciprocal functions, defined as the reciprocals of the main functions.

Main Function

  • Sine (sin)
  • Cosine (cos)
  • Tangent (tan)

Reciprocal Function

  • Cosecant (csc or cosec)
  • Secant (sec)
  • Cotangent (cot)

Angle A faces the side called the opposite side. It’s next to the side called the adjacent side. The remaining side is the hypotenuse, which is always the longest side in a right triangle.

1. The sine (abbreviated sin) is the ratio of the opposite side to the hypotenuse.

sin A = a/c

2. The cosine (abbreviated cos) is the ratio of the adjacent side to the hypotenuse.

cos A = b/c

3. The tangent (abbreviated tan) is the ratio of the opposite side to the adjacent side.

tan A = a/b

4. The cosecant (abbreviated csc) is a variation of the sine function. It’s the ratio of the hypotenuse to the opposite side.

csc A = c/a

5. The secant (abbreviated sec) is a variation of the cosine function. It’s the ratio of the hypotenuse to the adjacent side.

sec A = c/b

6. The cotangent (abbreviated cot) is a variation of the tangent function, but with the terms inverted. It’s the ratio of the adjacent side to the opposite side.

cot A = b/a

Law of Sines

The law of sines states that

a/Sin A = b/Sin B = c/Sin C

This law applies to any triangle with sides a, b, and c and angles A, B, and C, where A is the angle opposite side a, B is the angle opposite side b, and C is the angle opposite side c.

The value of the law of sines is that when you know three measurements of a triangle (as long as they include at least one side), you solve for the fourth measurement with no problem. You must have two angles and a side or two sides and an angle.