5.2 In die diagram is P(3 ; t) 'n punt in die Cartesiese vlak - NSC Mathematics - Question 5 - 2017 - Paper 2

The tangent of angle β can be defined as the ratio of the opposite side to the adjacent side in a right triangle. Thus, for angle β:

$\tan \beta = \frac{opposite}{adjacent} = \frac{3}{t}$

Substituting the value of $t$ found in the previous part:

$\tan \beta = \frac{3}{5}$

To find $\, \, cos \, 2β$, we can use the double angle formula for cosine:

$\cos 2β = 2\cos^2 \beta - 1$

First, we need to find $\cos \beta$. From the tangent value, we can derive:

$\cos \beta = \frac{adjacent}{hypotenuse} = \frac{t}{OP} = \frac{5}{\sqrt{34}}$

Now we can square this value:

$\cos^2 \beta = \left(\frac{5}{\sqrt{34}}\right)^2 = \frac{25}{34}$

Substituting into the double angle formula:

$\cos 2β = 2 \left(\frac{25}{34}\right) - 1 = \frac{50}{34} - 1 = \frac{50}{34} - \frac{34}{34} = \frac{16}{34} = \frac{8}{17}$