Trigonometric Ratio Of Special Angles

Explanation
The sine of 45 degrees is equal to the square root of 2 divided by 2. The sine function relates the ratio of the length of the side opposite to the angle to the length of the hypotenuse in a right triangle. In a 45-45-90 triangle, the two legs are congruent, and the hypotenuse is equal to the length of the leg multiplied by the square root of 2. Therefore, the length of the leg opposite the 45-degree angle is equal to the length of the hypotenuse divided by the square root of 2, which simplifies to the square root of 2 divided by 2.

Explanation
The sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right triangle. In a 30-60-90 triangle, the side opposite the 60-degree angle is equal to the square root of 3 times the length of the shorter leg, and the hypotenuse is twice the length of the shorter leg. Therefore, the sine of 60 degrees is equal to the square root of 3 divided by 2.

Explanation
The value of tan 45o is 1. In a right triangle, the tangent of an angle is equal to the length of the side opposite the angle divided by the length of the adjacent side. In a 45-45-90 triangle, the opposite side and the adjacent side are equal in length, so the tangent of 45 degrees is 1.

Explanation
The value of cos 45 degrees is √2/2. This can be determined by using the unit circle or by using the special right triangle with angles 45-45-90. In this triangle, the adjacent side and hypotenuse have the same length, which is √2, and the cosine of an angle is equal to the length of the adjacent side divided by the length of the hypotenuse. So, cos 45 degrees is √2/2.

Explanation
The sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right triangle. In a right triangle with a 30-degree angle, the side opposite the angle is equal to half the length of the hypotenuse. Therefore, the sine of 30 degrees is equal to 1/2.