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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)?