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3 years ago

3. Triangle ABC has angle measures as shown. a) What is the value of x? Show your work.

Mathematics

2 answers:

VladimirAG [237]3 years ago

8 0

As always, the sum of the measures of the angles in a triangle is 180°.

(x -5) +(3x +30) +35 = 180
4x +60 = 180 . . . . . . . . . . . . . collect terms
4x = 120 . . . . . . . . . . . . . . . . . subtract 60
x = 30 . . . . . . . . . . . . . . . . . divide by 4

a) The value of x is 30.

satela [25.4K]3 years ago

3 0

The sum of the angles in a triangle is 180°, therefore:
$(x - 5) + (3x + 30) + 35 = 180$
$x - 5 + 3x + 30 + 35 = 180$
Sum up x + 3x
$4x - 5 + 30 + 35 = 180$
Sum up all the numbers
$4x + 60 = 180$
Substract 60 from both sides
$4x = 120$
And finally divide by 4 (on both sides)
$x = 30$

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Given:

There are given that the cos function:

$cos210^{\circ}=-\frac{\sqrt{3}}{2}$

Explanation:

To find the value, first, we need to use the half-angle formula:

So,

From the half-angle formula:

$cos(\frac{\theta}{2})=\pm\sqrt{\frac{1+cos\theta}{2}}$

Then,

Since 105 degrees is the 2nd quadrant so cosine is negative

Then,

By the formula:

$\begin{gathered} cos(105^{\circ})=cos(\frac{210^{\circ}}{2}) \\ =-\sqrt{\frac{1+cos(210)}{2}} \end{gathered}$

Then,

Put the value of cos210 degrees into the above function:

So,

$\begin{gathered} cos(105^{\circ})=-\sqrt{\frac{1+cos(210)}{2}} \\ cos(105^{\operatorname{\circ}})=-\sqrt{\frac{1-\frac{\sqrt{3}}{2}}{2}} \\ cos(105^{\circ})=-\sqrt{\frac{2-\sqrt{3}}{4}} \\ cos(105^{\circ})=-\frac{\sqrt{2-\sqrt{3}}}{2} \end{gathered}$

Final answer:

Hence, the value of the cos(105) is shown below:

$cos(105^{\operatorname{\circ}})=-\frac{\sqrt{2-\sqrt{3}}}{2}$

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