Question

triangle lmn is a right triangle. suppose \\(\cos(\angle l) = 0.42\\).

which of these statements is true? for help, see this worked example.

\\(\sin(\angle n) = 0.58\\)
\\(\cos(\angle n) = 0.58\\)
\\(\sin(\angle n) = 0.42\\)
\\(\cos(\angle n) = 0.42\\)

Explanation:

Identify the given information

We are given a right triangle \(LMN\) with the right angle at vertex \(M\). We are also given that \(\cos(\angle L) = 0.42\).

Relate the acute angles

In any right triangle, the two acute angles are complementary. Since \(\angle M = 90^\circ\), we have:

$$ \angle L + \angle N = 90^\circ $$

Apply trigonometric definitions

Using the definitions of sine and cosine for the acute angles in right triangle \(LMN\):

$$ \cos(\angle L) = \frac{\text{Adjacent to } \angle L}{\text{Hypotenuse}} = \frac{ML}{LN} $$

$$ \sin(\angle N) = \frac{\text{Opposite to } \angle N}{\text{Hypotenuse}} = \frac{ML}{LN} $$

Equate the trigonometric ratios

Since both ratios equal \(\frac{ML}{LN}\), we establish the cofunction identity:

$$ \sin(\angle N) = \cos(\angle L) $$

Substitute the given value

Substitute the given value \(\cos(\angle L) = 0.42\) into the identity:

$$ \sin(\angle N) = 0.42 $$

Answer:

• \(\sin(\angle N) = 0.58\)
• \(\cos(\angle N) = 0.58\)
• \(\sin(\angle N) = 0.42\) (Correct answer)
• \(\cos(\angle N) = 0.42\)