Question

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

Answer 1

b)
`\sin(B) = (\sqrt 2)/(2)`

e)
`\tan(B) = 1`

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) = (Adjacent)/(Hypotenuse)`

This gives:

`\cos(B) = (a)/(c)`

Where:

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

`\cos(B) = (3)/(3\sqrt 2)`

3 cancels out

`\cos(B) = (1)/(\sqrt 2)`

Rationalize

`\cos(B) = (1)/(\sqrt 2)* (\sqrt 2)/(\sqrt 2)`

`\cos(B) = (\sqrt 2)/(\sqrt 2*\sqrt 2)`

`\cos(B) = (\sqrt 2)/(12)`

Also:

`\sin(B) = (Opposite)/(Hypotenuse)`

This gives:

`\sin(B) = (b)/(c)`

Where:

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

`\sin(B) = (3)/(3\sqrt 2)`

3 cancels out

`\sin(B) = (1)/(\sqrt 2)`

Rationalize

`\sin(B) = (1)/(\sqrt 2)* (\sqrt 2)/(\sqrt 2)`

`\sin(B) = (\sqrt 2)/(\sqrt 2 * \sqrt 2 )`

`\sin(B) = (\sqrt 2)/(2)`

Lastly:

`\tan(B) = (\sin(B))/(\cos(B))`

This gives:

`\tan(B) = (√(2)/2)/(√(2)/2)`

`\tan(B) = 1`

Hence, (b) and (e) are true