Question

given that cos0=sqrt2/5 of a triangle on the unit circle what is the sine ratio of the same triangle The options are a.sin0=1 b.sin0=1/2 c.sin0=sqrt 2/2 d.sin0= sqrt3/2

Answer 1

Which option is the closest

Answer 2

Are you sure

Answer 3

To determine the sine ratio given that \(\cos 0 = \frac{\sqrt{2}}{5}\), we can use the Pythagorean identity:

\[
\sin^2 \theta + \cos^2 \theta = 1
\]

First, we substitute \(\cos 0\) into the identity:

\[
\sin^2 0 + \left(\frac{\sqrt{2}}{5}\right)^2 = 1
\]

Calculating \(\left(\frac{\sqrt{2}}{5}\right)^2\):

\[
\left(\frac{\sqrt{2}}{5}\right)^2 = \frac{2}{25}
\]

Now, substituting this into the Pythagorean identity:

\[
\sin^2 0 + \frac{2}{25} = 1
\]

Now, isolate \(\sin^2 0\):

\[
\sin^2 0 = 1 - \frac{2}{25}
\]

We can express 1 as \(\frac{25}{25}\):

\[
\sin^2 0 = \frac{25}{25} - \frac{2}{25} = \frac{23}{25}
\]

Now, take the square root of both sides to find \(\sin 0\):

\[
\sin 0 = \sqrt{\frac{23}{25}} = \frac{\sqrt{23}}{5}
\]

None of the given options (a. \(1\), b. \(\frac{1}{2}\), c. \(\frac{\sqrt{2}}{2}\), d. \(\frac{\sqrt{3}}{2}\)) match \(\frac{\sqrt{23}}{5}\). Therefore, it appears that none of the provided options are correct based on the values given.

Answer 4

To find the closest option to \(\sin 0 = \frac{\sqrt{23}}{5}\), we first approximate \(\sqrt{23}\):

\[
\sqrt{23} \approx 4.79
\]

Now, we can calculate \(\frac{\sqrt{23}}{5}\):

\[
\frac{4.79}{5} \approx 0.958
\]

Now, let's evaluate the sine values from the given options:

a. \( \sin 0 = 1 \)

b. \( \sin 0 = \frac{1}{2} = 0.5 \)

c. \( \sin 0 = \frac{\sqrt{2}}{2} \approx 0.707 \)

d. \( \sin 0 = \frac{\sqrt{3}}{2} \approx 0.866 \)

Now, let's compare these approximate values:

- **Option a:** 1
- **Option b:** 0.5
- **Option c:** 0.707
- **Option d:** 0.866

Among the options provided, \(0.866\) (which corresponds to option d, \(\frac{\sqrt{3}}{2}\)) is the closest to \(0.958\), which is the approximate value of \(\sin 0 = \frac{\sqrt{23}}{5}\).

Therefore, the closest option is **d. \(\sin 0 = \frac{\sqrt{3}}{2}\)**.

Answer 5

Let's clarify this step by step.

We found that \(\sin 0 = \frac{\sqrt{23}}{5}\) and calculated an approximate value:

\[
\sqrt{23} \approx 4.79
\]

So,

\[
\sin 0 \approx \frac{4.79}{5} \approx 0.958
\]

Now, let's look again at the approximate values for the provided options:

1. **Option a:** \(\sin 0 = 1\) which is \(1.0\)
2. **Option b:** \(\sin 0 = \frac{1}{2} = 0.5\)
3. **Option c:** \(\sin 0 = \frac{\sqrt{2}}{2} \approx 0.707\)
4. **Option d:** \(\sin 0 = \frac{\sqrt{3}}{2} \approx 0.866\)

Now comparing these values:

- **Option a:** \(1.0\)
- **Option b:** \(0.5\)
- **Option c:** \(0.707\)
- **Option d:** \(0.866\)

The calculated sine value \(\approx 0.958\) is closer to **Option a (1)** than to option d (0.866), which is quite a bit lower.

So, upon reevaluation, the closest option to \(\sin 0 = \frac{\sqrt{23}}{5} \approx 0.958\) is actually **Option a: \(\sin 0 = 1\)**.

I apologize for my earlier oversight. Thank you for your patience!