A right triangle has a hypotenuse of length 3.00 m, and one of its angles is 30.0°. What are the lengths of (a) the side opposite the 30.0° angle and (b) the side adjacent to the 30.0° angle?

Well, let's analyze the situation. We have a right triangle with a hypotenuse of 3.00 m and one angle of 30.0°.

(a) To find the length of the side opposite the 30.0° angle, we can use the sine function. So let me reach deep into my mathematical humor closet and pull out this funny equation:

sin(30°) = Opposite / Hypotenuse

Now, if you're not allergic to math, we can plug in the values and simplify:

sin(30°) = Opposite / 3.00 m

Oh, wait! You may have forgotten to roll up your sleeves and break out your calculator. I don't blame you. Keep calm, and let's crunch the numbers. Sin(30°) is equal to 0.5. So we can rewrite the equation as:

0.5 = Opposite / 3.00 m

Now, to isolate the length of the side opposite the 30.0° angle, we'll multiply both sides of the equation by 3.00 m:

(0.5) * 3.00 m = Opposite

Finally, after a little bit of quick math:

Opposite = 1.50 m

So, the length of the side opposite the 30.0° angle is 1.50 meters.

(b) As for the side adjacent to the 30.0° angle, let's call it The Joker. Just kidding! Let's call it "Adjacent", because it's right there next to the angle!

To find the length of the side Adjacent, we'll use the cosine function:

cos(30°) = Adjacent / Hypotenuse

Remember, we do not need to go all the way to the cosine store to purchase an Adjacent. We already have it. We just need to find its length. So, diving right into the math:

cos(30°) = Adjacent / 3.00 m

Here's that magical number again: cos(30°) is equal to 0.866. Substitute it in:

0.866 = Adjacent / 3.00 m

Once again, let's isolate Adjacent by multiplying both sides by 3.00 m:

(0.866) * 3.00 m = Adjacent

After a bit of calculation:

Adjacent = 2.598 m

So the length of the side Adjacent to the 30.0° angle is approximately 2.598 meters.

Now, I hope you found my quirky math humor both enlightening and entertaining!

To find the lengths of the sides of the right triangle, we can use trigonometric ratios.

(a) The side opposite the 30.0° angle is referred to as the "opposite" side. We can find its length using the sine function.

sin(30.0°) = opposite / hypotenuse

Rearranging the equation, we have:

opposite = sin(30.0°) * hypotenuse

Substituting the given values, we have:

opposite = sin(30.0°) * 3.00 m

Using a calculator, we can find:

opposite ≈ 1.50 m

Therefore, the length of the side opposite the 30.0° angle is approximately 1.50 m.

(b) The side adjacent to the 30.0° angle is referred to as the "adjacent" side. We can find its length using the cosine function.

cos(30.0°) = adjacent / hypotenuse

Rearranging the equation, we have:

adjacent = cos(30.0°) * hypotenuse

Substituting the given values, we have:

adjacent = cos(30.0°) * 3.00 m

Using a calculator, we can find:

adjacent ≈ 2.60 m

Therefore, the length of the side adjacent to the 30.0° angle is approximately 2.60 m.

To find the lengths of the sides of a right triangle given the hypotenuse and one angle, we can use trigonometric ratios. In this case, we know the hypotenuse is 3.00 m and one of the angles is 30.0°.

(a) To find the length of the side opposite the 30.0° angle, we will use the sine ratio. The sine of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse.

sine ratio = opposite / hypotenuse

sin(30.0°) = opposite / 3.00 m

Now, we can rearrange the equation to solve for opposite:

opposite = sin(30.0°) * 3.00 m

Using a calculator, we find:

opposite = 1/2 * 3.00 m
opposite = 1.50 m

So, the length of the side opposite the 30.0° angle is 1.50 m.

(b) To find the length of the side adjacent to the 30.0° angle, we will use the cosine ratio. The cosine of an angle in a right triangle is defined as the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.

cosine ratio = adjacent / hypotenuse

cos(30.0°) = adjacent / 3.00 m

Now, we can rearrange the equation to solve for adjacent:

adjacent = cos(30.0°) * 3.00 m

Using a calculator, we find:

adjacent = √3/2 * 3.00 m
adjacent ≈ 2.598 m

So, the length of the side adjacent to the 30.0° angle is approximately 2.598 m.

a) sin 30º = opposite/3.00

opposite = 3.00(sin 30º) = ....

b) adjacent = 3.00(cos 30º) = ....