Answered

In a right-angled triangle [tex]\(GAP\)[/tex], [tex]\(AP = 12 \, \text{m}\)[/tex] and [tex]\(GA = 5 \, \text{m}\)[/tex]. The point [tex]\(T\)[/tex] on [tex]\(AP\)[/tex] is such that [tex]\(\angle AGT = \angle TGP = x^{\circ}\)[/tex]. Without using a calculator, find the exact values of the following:

a. [tex]\(\tan (2x)\)[/tex]

b. [tex]\(\tan x\)[/tex], by using the double angle formula

c. [tex]\(AT\)[/tex]



Answer :

GinnyAnswer
To solve the problem, let's start by recalling the given dimensions and using the properties of right-angled triangles and trigonometric identities.

Given:
- [tex]\( AP = 12 \)[/tex] meters
- [tex]\( GA = 5 \)[/tex] meters

First, we need to find the hypotenuse [tex]\( GP \)[/tex] of the right-angled triangle [tex]\( GAP \)[/tex].

### 1. Finding [tex]\( GP \)[/tex]:

Using the Pythagorean theorem, we have:
[tex]\[ GP = \sqrt{GA^2 + AP^2} \][/tex]
Substituting the given values:
[tex]\[ GP = \sqrt{5^2 + 12^2} \][/tex]
[tex]\[ GP = \sqrt{25 + 144} \][/tex]
[tex]\[ GP = \sqrt{169} \][/tex]
[tex]\[ GP = 13 \][/tex] meters

### 2. Trigonometric Ratios:

From the triangle [tex]\( GAP \)[/tex]:
[tex]\[ \cos x = \frac{GA}{GP} = \frac{5}{13} \][/tex]
[tex]\[ \sin x = \frac{AP}{GP} = \frac{12}{13} \][/tex]

### 3. Finding [tex]\( \tan x \)[/tex]:

[tex]\[ \tan x = \frac{\sin x}{\cos x} = \frac{\frac{12}{13}}{\frac{5}{13}} = \frac{12}{5} = 2.4 \][/tex]

### 4. Finding [tex]\( \tan (2x) \)[/tex]:

Using the double angle formula for tangent:
[tex]\[ \tan (2x) = \frac{2 \tan x}{1 - \tan^2 x} \][/tex]
Substitute [tex]\( \tan x = 2.4 \)[/tex]:
[tex]\[ \tan (2x) = \frac{2 \cdot 2.4}{1 - (2.4)^2} \][/tex]
[tex]\[ \tan (2x) = \frac{4.8}{1 - 5.76} \][/tex]
[tex]\[ \tan (2x) = \frac{4.8}{-4.76} \][/tex]
[tex]\[ \tan (2x) = -1.0084033613445378 \][/tex]

### 5. Finding [tex]\( AT \)[/tex]:

By the similarity of triangles [tex]\( TGA \)[/tex] and [tex]\( TGP \)[/tex], we have:
[tex]\[ \frac{AT}{AP} = \frac{GA}{GP} \][/tex]
[tex]\[ AT = \frac{GA \cdot AP}{GP} \][/tex]
Substitute the known values:
[tex]\[ AT = \frac{5 \cdot 12}{13} \][/tex]
[tex]\[ AT = \frac{60}{13} \][/tex]
[tex]\[ AT = 4.615384615384615 \][/tex] meters

### Summary:

a. [tex]\( \tan (2x) = -1.0084033613445378 \)[/tex]
b. [tex]\( \tan x = 2.4 \)[/tex]
c. [tex]\( AT = 4.615384615384615 \)[/tex] meters