Question

Дано три точки: O(0;0) A (1;0) B (1;1).Знайти синус і косинус кута AOB​

Answer 1

Ответ:

∠AOB=45°

Объяснение:

AO²=(1-0)²+(0-0)²=1+0=1⇒AO=1

BO²=(1-0)²+(1-0)²=1+1=2⇒BO=√2

AB²=(1-1)²+(0-1)²=0+1=1

AB²=AO²+BO²-2AO·BAcos∠AOB

1=1+2-2·1·√2cos∠AOB=3-2√2cos∠AOB

2√2cos∠AOB=3-1=2

cos∠AOB=√2/2⇒∠AOB=45°

Answer 2

Given Points and Angle Calculation

We are given three points: O(0,0), A(1,0), and B(1,1). We need to find the sine and cosine of the angle AOB.

To calculate the sine and cosine of the angle AOB, we can use the properties of vectors. The angle between two vectors can be calculated using the dot product and the magnitudes of the vectors.

Let's calculate the vectors OA and OB:

OA = A - O = (1, 0) - (0, 0) = (1, 0) OB = B - O = (1, 1) - (0, 0) = (1, 1)

The dot product of two vectors can be calculated by multiplying their corresponding components and summing them:

OA · OB = (1 * 1) + (0 * 1) = 1

The magnitudes of the vectors can be calculated using the Pythagorean theorem:

|OA| = sqrt((1^2) + (0^2)) = 1 |OB| = sqrt((1^2) + (1^2)) = sqrt(2)

Now, we can calculate the cosine of the angle AOB using the dot product and magnitudes:

cos(AOB) = (OA · OB) / (|OA| * |OB|) = 1 / (1 * sqrt(2)) = 1 / sqrt(2) = sqrt(2) / 2

To calculate the sine of the angle AOB, we can use the relationship between sine and cosine:

sin(AOB) = sqrt(1 - cos^2(AOB)) = sqrt(1 - (sqrt(2) / 2)^2) = sqrt(1 - 2 / 4) = sqrt(2 - 1) = sqrt(1) = 1

Therefore, the sine of the angle AOB is 1 and the cosine of the angle AOB is sqrt(2) / 2.

Please let me know if I can help you with anything else.