# Trigonometry

## Summary

- Trigonometry is the mathematical study of the relations of the magnitudes of the sides and angles of right triangles (triangles with with one 90° angle).
- Parts of triangles
- The location of the sides of any triangle are identified relative to an angle using the words opposite and adjacent.
- The sides of a right triangle adjacent to the right angle are called legs.
- The side of a right triangle opposite the right angle is called the hypotenuse.

- Standard position notation is used in this book
- Start with a diagram of a rectangular coordinate system.
- Label the "horizontal" axis pointing to the right of the diagram +
*x*. - Label the "vertical" axis pointing to the top of the diagram +
*y*.

- Label the "horizontal" axis pointing to the right of the diagram +
- Take any right triangle with one angle θ.
- Place the vertex of the angle θ on the origin.
- Align the leg adjacent to θ on the +
*x*axis.

- Label the sides of the triangle.
*x*is the leg adjacent to the angle (usually shortened to the adjacent).*y*is the leg opposite the angle (usually shortened to the opposite).*R*is the hypotenuse.

- The symbol
*R*(or*r*) is used because the hypotenuse can be thought of…- as the
*resultant*formed by the addition*x*of and*y*. - as the
*radius*of a circle centered on the origin.

- as the

- Start with a diagram of a rectangular coordinate system.
- Triangle inequality
- For any triangle, the sum of the lengths of any two sides is greater than or equal to the length of the remaining side.
- For a right triangle, the sum of the lengths of the legs is greater than or equal to the length of the hypotenuse.
*x*+*y*≥*R*

- Pythagorean theorem
- For a right triangle, the square of the hypotenuse is equal to the the sum of the squares of the legs.
*R*^{2}=*x*^{2}+*y*^{2} - The hypotenuse is always the longest side of a right triangle.

- For a right triangle, the square of the hypotenuse is equal to the the sum of the squares of the legs.
- Trigonometric functions…
- are ratios of the lengths of the sides of a right triangle
- as ratios, they are unitless (dimensionless)

- are defined relative to one of the angles that is not 90°
- often given the generic symbol θ (theta)

- come in six types, three of which are commonly used in physics…
Commonly used trigonometric functions sine cosine tangent sin θ = *opposite*= *y**hypotenuse**R*cos θ = *adjacent*= *x**hypotenuse**R*tan θ = *opposite*= *y**adjacent**x*

- are ratios of the lengths of the sides of a right triangle