Answer :

msm555

Answer:

[tex]\bold{\sf x = \boxed{33.3^\circ} }[/tex]

Step-by-step explanation:

Given:

  • [tex]\bold{\sf N = x^\circ }[/tex],
  • [tex]\bold{\sf NO = 51 }[/tex], and. [tex]\bold{\sf PN = 61 }[/tex]

To find:

x = ?

Solution:

To solve for [tex]\bold{\sf x }[/tex] in triangle [tex]\bold{\sf NOP }[/tex], we can use the cosine ratio in a right triangle.

The cosine ratio is given by:

[tex]\large\boxed{\boxed{\sf \cos(N) = \dfrac{\textsf{adjacent}}{\textsf{hypotenuse}}}} [/tex]

In this case, the adjacent side to angle [tex]\bold{\sf N }[/tex] is [tex]\bold{\sf NO }[/tex], and the hypotenuse is [tex]\bold{\sf PN }[/tex].

So, we have:

[tex]\sf \cos(N) = \dfrac{NO}{PN} [/tex]

Substitute the given values:

[tex]\sf \cos(x^\circ) = \dfrac{51}{61} [/tex]

Now, to solve for [tex]\bold{\sf x }[/tex], we need to take the inverse cosine (also known as arccosine) of both sides:

[tex]\sf x^\circ = \cos^{-1} \left(\dfrac{51}{61}\right) [/tex]

[tex]\sf x^\circ \approx \cos^{-1} \left(0.8360655737704\right) [/tex]

Using a calculator, we find:

[tex]\sf x^\circ \approx 33.273043412736^\circ [/tex]

[tex]\sf x^\circ \approx 33.3^\circ\textsf{ (in nearest tenth)} [/tex]

Therefore, [tex]\bold{\sf x \approx \boxed{33.3^\circ} }[/tex].

semsee45

Answer:

x = 33.3

Step-by-step explanation:

The diagram shows right triangle PON, with its right angle at vertex O. The angle at vertex N is labelled 'x', the side adjacent this angle measures 51 units, and the hypotenuse measures 61 units.

To find the value of x, we can use the cosine trigonometric ratio, since we have been given the measures of the hypotenuse and the side adjacent the angle.

[tex]\boxed{\begin{array}{l}\underline{\textsf{Cosine trigonometric ratio}}\\\\\sf \cos \theta=\dfrac{A}{H}\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$\theta$ is the angle.}\\\phantom{ww}\bullet\;\textsf{$A$ is the side adjacent the angle.}\\\phantom{ww}\bullet\;\textsf{$H$ is the hypotenuse (the side opposite the right angle).}\end{array}}[/tex]

In this case:

  • θ = x
  • A = 51
  • H = 61

Substitute the values into the cosine ratio and solve for x:

[tex]\cos x^{\circ}=\dfrac{51}{61}\\\\\\x^{\circ}=\cos^{-1}\left(\dfrac{51}{61}\right)\\\\\\x^{\circ}=33.27304341...^{\circ}\\\\\\x=33.3\; \sf (nearest\;tenth)[/tex]

Therefore, the value of x rounded to the nearest tenth is:

[tex]\Large\boxed{\boxed{x=33.3}}[/tex]

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