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.
  • 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…

• Special triangle relationships
  • Some right triangles have side-lengths that are related by simple ratios. Knowing this can be helpful for solving some problems using trigonometry.