Question

Which statements are true about angle b in this triangle?​

Answer 1

Answer:

b) $\sin(B) = \frac{\sqrt 2}{2}$

e) $\tan(B) = 1$

Step-by-step explanation:

Given

The attached triangle

Required

The true statement about B

The given options show that we determine the correct trigonometry ratio of B.

So, we have:

$\cos(B) = \frac{Adjacent}{Hypotenuse}$

This gives:

$\cos(B) = \frac{a}{c}$

Where:

$a = 3mm\\c = 3\sqrt 2 mm$

$\cos(B) = \frac{3}{3\sqrt 2}$

3 cancels out

$\cos(B) = \frac{1}{\sqrt 2}$

Rationalize

$\cos(B) = \frac{1}{\sqrt 2}* \frac{\sqrt 2}{\sqrt 2}$

$\cos(B) = \frac{\sqrt 2}{\sqrt 2*\sqrt 2}$

$\cos(B) = \frac{\sqrt 2}{12}$

Also:

$\sin(B) = \frac{Opposite}{Hypotenuse}$

This gives:

$\sin(B) = \frac{b}{c}$

Where:

$b = 3mm\\c = 3\sqrt 2 mm$

$\sin(B) = \frac{3}{3\sqrt 2}$

3 cancels out

$\sin(B) = \frac{1}{\sqrt 2}$

Rationalize

$\sin(B) = \frac{1}{\sqrt 2}* \frac{\sqrt 2}{\sqrt 2}$

$\sin(B) = \frac{\sqrt 2}{\sqrt 2 * \sqrt 2 }$

$\sin(B) = \frac{\sqrt 2}{2}$

Lastly:

$\tan(B) = \frac{\sin(B)}{\cos(B)}$

This gives:

$\tan(B) = \frac{\sqrt{2}/2}{\sqrt{2}/2}$

$\tan(B) = 1$

Hence, (b) and (e) are true