Britannica Sources used in this answer

What are the exact sine, cosine, and tangent values for 0°, 30°, 45°, 60°, and 90°?

Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.

Diagram of a right triangle labeled with sides a, b, and hypotenuse c, alongside formulas for the six trigonometric functions-sine, cosine, tangent, cosecant, secant, and cotangent-each defined by ratios of the triangle's sides. — trigonometry Based on the definitions, various simple relationships exist among the functions. For example, csc A = 1/sin A, sec A = 1/cos A, cot A = 1/tan A, and tan A = sin A/cos A.

The exact values for the sine, cosine, and tangent functions for the specified angles are determined by geometric principles. These values are summarized in the table below:1

| Angle ( $θ$ ) | Sine ( $\sin θ$ ) | Cosine ( $\cos θ$ ) | Tangent ( $\tan θ$ ) |
| :--- | :--- | :--- | :--- |
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | $\sqrt{3} / 2$ | $\sqrt{3} / 3$ |
| 45° | $\sqrt{2} / 2$ | $\sqrt{2} / 2$ | 1 |
| 60° | $\sqrt{3} / 2$ | 1/2 | $\sqrt{3}$ |
| 90° | 1 | 0 | Undefined |

In trigonometry, these trigonometric functions represent ratios of the sides of a right triangle. For instance, the tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side, which is also equivalent to the ratio of the sine to the cosine of that angle. At 90°, the tangent is undefined because the cosine of 90° is 0, and division by zero is not defined in this context.234