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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

5 answers

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.

Which option is the closest

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

Are you sure

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!