In right triangle LMN, the right angle is at vertex M. The hypotenuse LN has a length of 2… — ACT

In right triangle $LMN$ , the right angle is at vertex $M$ . The hypotenuse $LN$ has a length of $25$ units. If $\cos(L) = \frac{7}{25}$ , what is the value of $\tan(N)$ ?

$\frac{7}{24}$ Answer
B
$\frac{24}{7}$
C
$\frac{7}{25}$
D
$\frac{24}{25}$
E
$\frac{25}{7}$

Answer

\frac{7}{24}

The ratio

\cos(L) = \frac{7}{25}

means the leg adjacent to angle

L

(

LM

) is

7

units and the hypotenuse (

LN

) is

25

units. Using the Pythagorean theorem (

7^2 + MN^2 = 25^2

), the remaining leg

MN

24

units. For angle

N

, the opposite side is

LM = 7

and the adjacent side is

MN = 24

. Thus,

\tan(N) = \frac{\text{opposite}}{\text{adjacent}} = \frac{7}{24}

Step-by-Step Solution

Use the definition of cosine for angle

L

to identify the length of leg

LM

Since

\cos(L) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{LM}{25} = \frac{7}{25}

, the length of leg

LM

7

units.

Cosine is defined as the ratio of the adjacent side to the hypotenuse in a right triangle.

Calculate the length of the remaining leg

MN

using the Pythagorean theorem.

MN = \sqrt{LN^2 - LM^2} = \sqrt{25^2 - 7^2} = \sqrt{625 - 49} = \sqrt{576} = 24

units.

In a right triangle, the square of the hypotenuse equals the sum of the squares of the legs.

Set up the tangent ratio for angle

N

\tan(N) = \frac{\text{opposite side to } N}{\text{adjacent side to } N} = \frac{LM}{MN} = \frac{7}{24}

Tangent is defined as the ratio of the side opposite the angle to the side adjacent to the angle.

Key Concept

Right Triangle Trigonometric Ratios (SOHCAHTOA)

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