Home > Do Mathematics -> Trigonometry -> The Tangent Function
 Definition of tangent
Given a right-triangle (a triangle with a right angle) with acute angle α, the tangent of α is the value of the ratio: $ \tan(\alpha)=\large{\frac{\text{side opposite}\,\, \alpha}{\text{side adjacent}\,\, \alpha}}=\large{\frac{\text{opp}}{\text{adj}}} $

 Definition of tangent    Directions and Links
Link to GeoGebraTube:   Teacher   Student
  • Click and drag point A or point B to change the triangle. (Angle ACB is always a right 90° angle so C depends on A and B).
  • Find the value of tan(α) using your calculator. Use either
    • the tan key and the size of the angle α or
    • just divide the sides to find the value of the fraction $\frac{a}{b} $
  • (Both ways of calculating should give you the same value.)

 Tangent - Nested Triangles    Directions and Links
Link to GeoGebraTube:   Teacher   Student
  • Click and drag the angle α to any value between 0° and 90°.
  • Then click and drag b and b' to make any 2 right-triangles you want with angle α.
  • Look at first triangle: Divide the length of the side opposite to the angle α by the length of the side adjacent to the angle α. Write this number down.
  • Look at the second triangle and again divide the length of the side opposite to the angle α by the length of the side adjacent to the angle α. It should be the same number!
  • This is the value of tan(α).

 y=tan(x) - Degrees    Directions and Links
Link to GeoGebraTube:   Teacher   Student
  • Move the slider to α=30°. The tan(30°)≈0.577. Do you know the exact value of tan(30°)?
  • Move the slider to α=120°. The tan(120°)≈ –1.732. Do you know the exact value of tan(120°)?
  • Now, let α=–60°. Move the slider to the corresponding angle between 0° and 360°.
  • The tan(–30°) = tan(330°) ≈ –0.577 . Do you know the exact value of tan(–30°)?
  • Does this agree with the fact that tan(–α) = –tan(α)?

 y=tan(x) - Radians  Directions and Links
Link to GeoGebraTube:   Teacher   Student
  • First notice the unit circle has a radius of 1.
  • Click and drag the slider for α. Note the size of α in degrees and in radians. (Although we wrote the word radians, it should NOT be written. There is no unit with radians.)
  • The ANGLE α is on the x-axis. The ANGLE α is the LENGTH OF THE ARC in the unit circle for the angle α in degrees. Read this sentence until you understand it. It is critical.
  • The value of the tangent function of the angle α is the y-coordinate of the point T divided by the x-coordinate of the point T.
  • Notice that from 0 to π/4; the height is less than the width and tangent is less than 1.
  • Notice that from π/4 to π/2; the height is greater than the width and tangent is greater than 1.
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