Explain how to use the measures of a right triangle to calculate the exact value of sin 30 degrees? How can this information be used to determine the exact value of sin 60 degrees?

You know the sides are in the ratio 1:√3:2

Just switch angles to figure sin60

To calculate the exact value of sin 30 degrees, we can use the measures of a right triangle. First, let's draw a right triangle and label the angles and sides. The angle opposite the 30-degree angle will be 90 - 30 = 60 degrees. The side opposite the 30-degree angle will be the height of the triangle, while the side opposite the 60-degree angle will be the base.

Now, we need to determine the lengths of the sides of the triangle. Let's assume the hypotenuse is 2 units. Since it's a right triangle, we can use the Pythagorean theorem to find the length of the remaining side.

According to the Pythagorean theorem:
c^2 = a^2 + b^2

In this case, c represents the hypotenuse, a represents the height, and b represents the base.

Therefore, 2^2 = a^2 + b^2
4 = a^2 + b^2

Now, let's find the length of the height (a). Since the angle opposite the height is 30 degrees, and the hypotenuse is 2, we can use the sin function to find the length of the height.

sin 30 = a / c
a = sin 30 * c
a = sin 30 * 2
a = 1 * 2
a = 2

So, the length of the height is 2 units.

To find the length of the base (b), we need to use the Pythagorean theorem:

4 = 2^2 + b^2
4 = 4 + b^2
b^2 = 0
b = 0

Therefore, the length of the base is 0 units.

Now, we can use the ratio of the opposite side (height) to the hypotenuse to determine the value of sin 30 degrees.

sin 30 = a / c
sin 30 = 2 / 2
sin 30 = 1

So, the exact value of sin 30 degrees is 1.

Now, to determine the exact value of sin 60 degrees, we can use the same triangle. We already know that the length of the height (opposite the 30-degree angle) is 2 units.

The angle opposite the 60-degree angle will be 180 - 90 - 60 = 30 degrees. So, the side opposite the 60-degree angle will be the height of the triangle, which is also 2 units.

To find the length of the base, we can use the Pythagorean theorem:

c^2 = a^2 + b^2
2^2 = 2^2 + b^2
4 = 4 + b^2
b^2 = 0
b = 0

Therefore, the length of the base is 0 units.

Now, we can use the ratio of the opposite side (height) to the hypotenuse to determine the value of sin 60 degrees.

sin 60 = a / c
sin 60 = 2 / 2
sin 60 = 1

So, the exact value of sin 60 degrees is also 1.