The Physics
Hypertextbook
Opus in profectus

# 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
• Label the "horizontal" axis pointing to the right of the diagram +x.
• Label the "vertical" axis pointing to the top of the diagram +y.
• 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.
• 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.

R2 = x2 + y2

• The hypotenuse is always the longest side of a right triangle.
• 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