A 25-foot ladder leans against a vertical wall on flat, horizontal ground. The base of the… — ACT

A 25-foot ladder leans against a vertical wall on flat, horizontal ground. The base of the ladder is positioned 7 feet away from the wall. If $\theta$ represents the measure of the angle formed between the ladder and the ground, what is the value of $\sin(\theta) + \cos(\theta)$ ?

$\frac{31}{25}$ Answer
B
$\frac{17}{25}$
C
$\frac{31}{24}$
D
$\frac{31}{7}$
E
$\frac{31}{\sqrt{674}}$

Answer

\frac{31}{25}

The ladder forms a 7-24-25 right triangle with the ground and the wall. Relative to angle

\theta

between the ladder and the ground, the opposite side is 24 feet, the adjacent side is 7 feet, and the hypotenuse is 25 feet. Evaluating

\sin(\theta) = \frac{24}{25}

and

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

and adding them gives

\frac{31}{25}

Step-by-Step Solution

Identify known components of the right triangle

The ladder acts as the hypotenuse (

c = 25

), and the distance along the ground is the adjacent side to angle

\theta

(

a = 7

The ladder, wall, and ground form a right triangle with the right angle at the base of the wall.

Calculate the height of the wall (opposite side)

Opposite side

b = \sqrt{25^2 - 7^2} = \sqrt{625 - 49} = \sqrt{576} = 24

feet.

By the Pythagorean theorem (

a^2 + b^2 = c^2

), solving for

b

gives

b = \sqrt{c^2 - a^2}

Determine

\sin(\theta)

and

\cos(\theta)

using SOHCAHTOA ratios

\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{24}{25}

and

\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{7}{25}

Sine is defined as opposite over hypotenuse, and cosine is defined as adjacent over hypotenuse.

Sum the trigonometric ratios

\sin(\theta) + \cos(\theta) = \frac{24}{25} + \frac{7}{25} = \frac{31}{25}

Add the two fractions with the common denominator 25.

Key Concept

Right Triangle Trigonometry (SOHCAHTOA) and Pythagorean Theorem

Estimated Time:1m 15s

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