Question

1) sin 765° 2) cos 420°

Answer 1

1)sin(765°)=sin(45°+2×360°)=sin(45°)=√2/2
2)cos(420°)=cos(60°+360°)=cos(60°)=1/2

Answer 2

1) Calculating sin(765°): To find the value of sin(765°), we need to convert 765° to an angle within the range of 0 to 360 degrees. We can do this by subtracting multiples of 360 from 765 until we get a value within that range. In this case, we have:

765° - 2 * 360° = 45°

Now we can calculate sin(45°). The sine function gives us the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right-angled triangle. In a unit circle, the hypotenuse is always 1.

For a 45° angle, we have an isosceles right-angled triangle with two sides of length 1. By the Pythagorean theorem, the length of the third side (opposite the right angle) is sqrt(2).

So, sin(45°) = opposite / hypotenuse = sqrt(2) / 1 = sqrt(2).

Therefore, sin(765°) = sin(45°) = sqrt(2).

2) Calculating cos(420°): To find the value of cos(420°), we follow a similar process. We convert 420° to an angle within the range of 0 to 360 degrees:

420° - 1 * 360° = 60°

Now we can calculate cos(60°). The cosine function gives us the ratio of the length of the side adjacent to the angle to the length of the hypotenuse in a right-angled triangle.

For a 60° angle, we have an equilateral triangle with all sides of length 1. By splitting the equilateral triangle in half, we get a right-angled triangle with a 30° angle. The adjacent side in this triangle has a length of 1/2, and the hypotenuse is 1.

So, cos(60°) = adjacent / hypotenuse = 1/2 / 1 = 1/2.

Therefore, cos(420°) = cos(60°) = 1/2.

In summary: 1) sin(765°) = sqrt(2) 2) cos(420°) = 1/2