Trigonometric Functions and Values

Trigonometric Functions and Their Values

Fundamental definitions

• Sine function (sin):

  • Definition: For an angle 𝜃, the sine function is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right triangle.

  • Formula: $ext{sin } heta = rac{y}{r}$ (where ( y ) = opposite side, ( r ) = hypotenuse)

• Cosine function (cos):

  • Definition: For an angle 𝜃, the cosine function is defined as the ratio of the length of the adjacent side to the hypotenuse in a right triangle.

  • Formula: $ext{cos } heta = rac{x}{r}$ (where ( x ) = adjacent side, ( r ) = hypotenuse)

• Tangent function (tan):

  • Definition: For an angle 𝜃, the tangent function is defined as the ratio of the sine to the cosine, or equivalently, the opposite side to the adjacent side in a right triangle.

  • Formula: $ext{tan } heta = rac{y}{x}$ (where ( y ) = opposite side, ( x ) = adjacent side)

Values of trigonometric functions at key angles

1. Angle $0$ radians (also known as $0^ ext{o}$):

   • $ext{sin } 0 = 0$

   • $ext{cos } 0 = 1$

   • $ext{tan } 0 = 0$

2. Angle $ rac{ ext{π}}{6}$ radians (or $30^ ext{o}$):

   • $ext{sin } rac{ ext{π}}{6} = rac{1}{2}$

   • $ext{cos } rac{ ext{π}}{6} = rac{ ext{√3}}{2}$

   • $ext{tan } rac{ ext{π}}{6} = rac{1}{ ext{√3}}$

3. Angle $ rac{ ext{π}}{4}$ radians (or $45^ ext{o}$):

   • $ext{sin } rac{ ext{π}}{4} = rac{ ext{√2}}{2}$

   • $ext{cos } rac{ ext{π}}{4} = rac{ ext{√2}}{2}$

   • $ext{tan } rac{ ext{π}}{4} = 1$

4. Angle $ rac{ ext{π}}{3}$ radians (or $60^ ext{o}$):

   • $ext{sin } rac{ ext{π}}{3} = rac{ ext{√3}}{2}$

   • $ext{cos } rac{ ext{π}}{3} = rac{1}{2}$

   • $ext{tan } rac{ ext{π}}{3} = ext{√3}$

5. Angle $ rac{ ext{π}}{2}$ radians (or $90^ ext{o}$):

   • $ext{sin } rac{ ext{π}}{2} = 1$

   • $ext{cos } rac{ ext{π}}{2} = 0$

   • $ext{tan } rac{ ext{π}}{2} = ext{undefined}$

6. Angle $ rac{2 ext{π}}{3}$ radians (or $120^ ext{o}$):

   • $ext{sin } rac{2 ext{π}}{3} = rac{ ext{√3}}{2}$

   • $ext{cos } rac{2 ext{π}}{3} = - rac{1}{2}$

   • $ext{tan } rac{2 ext{π}}{3} = - ext{√3}$

7. Angle $ rac{3 ext{π}}{4}$ radians (or $135^ ext{o}$):

   • $ext{sin } rac{3 ext{π}}{4} = rac{ ext{√2}}{2}$

   • $ext{cos } rac{3 ext{π}}{4} = - rac{ ext{√2}}{2}$

   • $ext{tan } rac{3 ext{π}}{4} = -1$

8. Angle $ rac{5 ext{π}}{6}$ radians (or $150^ ext{o}$):

   • $ext{sin } rac{5 ext{π}}{6} = rac{1}{2}$

   • $ext{cos } rac{5 ext{π}}{6} = - rac{ ext{√3}}{2}$

   • $ext{tan } rac{5 ext{π}}{6} = - rac{1}{ ext{√3}}$

9. Angle $ ext{π}$ radians (or $180^ ext{o}$):

   • $ext{sin } ext{π} = 0$

   • $ext{cos } ext{π} = -1$

   • $ext{tan } ext{π} = 0$

10. Angle $ rac{7 ext{π}}{6}$ radians (or $210^ ext{o}$):

    • $ext{sin } rac{7 ext{π}}{6} = - rac{1}{2}$

    • $ext{cos } rac{7 ext{π}}{6} = - rac{ ext{√3}}{2}$

    • $ext{tan } rac{7 ext{π}}{6} = rac{1}{ ext{√3}}$

11. Angle $ rac{5 ext{π}}{4}$ radians (or $225^ ext{o}$):

    • $ext{sin } rac{5 ext{π}}{4} = - rac{ ext{√2}}{2}$

    • $ext{cos } rac{5 ext{π}}{4} = - rac{ ext{√2}}{2}$

    • $ext{tan } rac{5 ext{π}}{4} = 1$

12. Angle $ rac{4 ext{π}}{3}$ radians (or $240^ ext{o}$):

    • $ext{sin } rac{4 ext{π}}{3} = - rac{ ext{√3}}{2}$

    • $ext{cos } rac{4 ext{π}}{3} = - rac{1}{2}$

    • $ext{tan } rac{4 ext{π}}{3} = ext{√3}$

13. Angle $ rac{3 ext{π}}{2}$ radians (or $270^ ext{o}$):

    • $ext{sin } rac{3 ext{π}}{2} = -1$

    • $ext{cos } rac{3 ext{π}}{2} = 0$

    • $ext{tan } rac{3 ext{π}}{2} = ext{undefined}$

14. Angle $ rac{5 ext{π}}{3}$ radians (or $300^ ext{o}$):

    • $ext{sin } rac{5 ext{π}}{3} = - rac{ ext{√3}}{2}$

    • $ext{cos } rac{5 ext{π}}{3} = rac{1}{2}$

    • $ext{tan } rac{5 ext{π}}{3} = - ext{√3}$

15. Angle $ rac{7 ext{π}}{4}$ radians (or $315^ ext{o}$):

    • $ext{sin } rac{7 ext{π}}{4} = - rac{ ext{√2}}{2}$

    • $ext{cos } rac{7 ext{π}}{4} = rac{ ext{√2}}{2}$

    • $ext{tan } rac{7 ext{π}}{4} = -1$

16. Angle $ rac{11 ext{π}}{6}$ radians (or $330^ ext{o}$):

    • $ext{sin } rac{11 ext{π}}{6} = - rac{1}{2}$

    • $ext{cos } rac{11 ext{π}}{6} = rac{ ext{√3}}{2}$

    • $ext{tan } rac{11 ext{π}}{6} = - rac{1}{ ext{√3}}$

Summary of trigonometric values

• These sine, cosine, and tangent values at key angles are crucial for solving many problems in trigonometry, calculus, and physics.