Why the Tangent Function Can be Greater Than 1, But the Sine and Cosine Functions Cannot

Regardless of the angle for which you evaluate tangent, cosine, or sine, you can always think of it as the ratio of two sides of a right triangle. See the orange angle (quadrant 1), green angle (quadrant 2), purple angle (quadrant 3), and red angle (quadrant 4) in the diagram.

For the tangent function, you can divide the opposite side by the adjacent side.

For the sine function, you can divide the opposite side by the hypotenuse.

For the cosine function, you can divide the adjacent side by the hypotenuse.

Three sides for Right Triangle

H: hypotenuse, O: opposite, A: Adjacent

For the three sides of a triangle, the two sides that are not the hypotenuse (opposite & adjacent) will never be larger than the hypotenuse. They can be equal to the hypotenuse, so you can get results of the sine and cosine functions being equal to one.

However, the opposite side of a triangle can definitely be larger than the adjacent side of a triangle.

tangent vs sin and cos ratios

In the triangle with green letters O, H, A the opposite and adjacent sides are similar with the opposite side being a bit larger.

In the triangle with blue letters O, H, A, the opposite side is significantly larger than the adjacent side.

In the third orange triangle, the opposite side is smaller than the adjacent side.

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