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}}} $

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!

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.

Definition of tangent Directions and Links

tankey and the size of the angle α orTangent - Nested Triangles Directions and Links

y=tan(x) - Degrees Directions and Linksy=tan(x) - Radians Directions and Links_{4}; the height is less than the width and tangent is less than 1._{4}to π/_{2}; the height is greater than the width and tangent is greater than 1.Related pages: