Question

l, m, n with triangle, 26, 10, right angle at n.
acent, and opposite.
sin(l) = 5/26 = 5/13
cos(l) = 24/26 = 12/13
tan(l) = 5/24 = 5/12

Explanation:

Step1: Identify Triangle Type

The triangle \( \triangle LNM \) has a right angle at \( N \), so it's a right - triangle. In a right - triangle, for an acute angle \( \theta \), \( \sin\theta=\frac{\text{opposite}}{\text{hypotenuse}} \), \( \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} \), and \( \tan\theta=\frac{\text{opposite}}{\text{adjacent}} \).

First, we need to find the length of \( LN \). Using the Pythagorean theorem in \( \triangle LNM \) (where \( LM = 26 \), \( MN=10 \)): Let \( LN=x \), then \( x^{2}+10^{2}=26^{2} \). So \( x^{2}=26^{2}- 10^{2}=(26 + 10)(26 - 10)=36\times16 = 576 \), so \( x = 24 \).

Step2: Analyze \( \sin(L) \)

For angle \( L \), the opposite side to angle \( L \) is \( MN = 10\div2 = 5\)? Wait, no, \( MN = 10 \), but in the formula \( \sin(L)=\frac{5}{26} \), maybe there is a miscalculation in the original problem or maybe \( MN \) is split? Wait, looking at the triangle, \( MN = 10 \), \( LN = 24 \), \( LM = 26 \). Wait, the opposite side to angle \( L \) should be \( MN \), adjacent is \( LN \), hypotenuse is \( LM \). But in the given \( \sin(L)=\frac{5}{26} \), maybe there is a typo and \( MN = 10 \) is actually \( 5\times2 \), so the opposite side is \( 5 \) (if we consider a smaller triangle or a ratio). But let's go with the given calculations.

For \( \sin(L) \): \( \sin(L)=\frac{\text{opposite}}{\text{hypotenuse}} \). If \( \sin(L)=\frac{5}{26}=\frac{5}{13\times2}=\frac{5}{13} \) (simplified), the opposite side is \( 5 \), hypotenuse is \( 26 \).

For \( \cos(L) \): \( \cos(L)=\frac{\text{adjacent}}{\text{hypotenuse}} \). The adjacent side to angle \( L \) is \( LN = 24 \), hypotenuse is \( 26 \). So \( \cos(L)=\frac{24}{26}=\frac{12}{13} \) (simplified by dividing numerator and denominator by 2).

For \( \tan(L) \): \( \tan(L)=\frac{\text{opposite}}{\text{adjacent}} \). The opposite side is \( 5 \) (if we take \( MN = 10 \) as \( 2\times5 \)) and adjacent is \( 24 \), so \( \tan(L)=\frac{5}{24}=\frac{5}{12} \) (simplified? Wait, \( 5/24 \) can't be simplified to \( 5/12 \) unless there is a mistake. Wait, maybe the opposite side is \( 10 \) and adjacent is \( 24 \), then \( \tan(L)=\frac{10}{24}=\frac{5}{12} \). Ah, that makes sense. So the opposite side is \( MN = 10 \), adjacent is \( LN = 24 \), so \( \tan(L)=\frac{10}{24}=\frac{5}{12} \), \( \sin(L)=\frac{10}{26}=\frac{5}{13} \), \( \cos(L)=\frac{24}{26}=\frac{12}{13} \).

Answer:

The trigonometric ratios for angle \( L \) in right - triangle \( LNM \) are \( \sin(L)=\frac{5}{13} \), \( \cos(L)=\frac{12}{13} \), \( \tan(L)=\frac{5}{12} \) (after correcting the possible mis - labeling of the opposite side as \( 5 \) instead of \( 10 \), probably a simplification by dividing \( 10 \) and \( 26 \) by \( 2 \) for \( \sin \), \( 24 \) and \( 26 \) by \( 2 \) for \( \cos \), and \( 10 \) and \( 24 \) by \( 2 \) for \( \tan \)).