Problem 14 - Algebra 2 and Trigonometry

Question

A 20-foot ladder leaning against a vertical wall reaches to a height of 16 feet. Find the sine, cosine, and tangent values of the angle that the ladder makes with the ground.

Step-by-Step Solution

Answer

\( \sin(\theta) = \frac{4}{5}, \cos(\theta) = \frac{3}{5}, \tan(\theta) = \frac{4}{3}. \)

Step 1: Identify the triangle

The ladder, the wall, and the ground form a right triangle. The ladder is the hypotenuse with a length of 20 feet, and the height the ladder reaches on the wall is one leg of the triangle, 16 feet.

Step 2: Determine the length of the second leg

Use the Pythagorean theorem to find the length of the second leg (ground distance). The formula is \[ c^2 = a^2 + b^2 \]where \( c = 20 \) and \( a = 16 \). Solve for \( b \): \[ b^2 = 20^2 - 16^2 \]\[ b^2 = 400 - 256 \]\[ b^2 = 144 \]\[ b = \sqrt{144} = 12 \]So, the length of the ground leg is 12 feet.

Step 3: Find the sine of the angle

The sine of the angle \( \theta \) is given by the ratio of the opposite side to the hypotenuse. \[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{16}{20} = \frac{4}{5} \]

Step 4: Find the cosine of the angle

The cosine of the angle \( \theta \) is given by the ratio of the adjacent side to the hypotenuse. \[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{12}{20} = \frac{3}{5} \]

Step 5: Find the tangent of the angle

The tangent of the angle \( \theta \) is given by the ratio of the opposite side to the adjacent side. \[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{16}{12} = \frac{4}{3} \]

Key Concepts

Right Triangle

A right triangle is a special type of triangle where one of its angles is exactly 90 degrees. This type of triangle is essential in trigonometry due to its unique properties. In a right triangle:

• The side opposite the right angle is called the hypotenuse, which is always the longest side.
• The other two sides are known as the legs of the triangle. They can be referred to as the opposite and adjacent sides depending on the context of the angle being examined.

In the problem scenario, the ladder serves as the hypotenuse, the wall-side leg is the opposite side with a length of 16 feet, and the ground distance (the other leg) is the adjacent side measuring 12 feet.
Understanding right triangles is crucial because it helps in calculating unknown sides or angles using trigonometric ratios and the Pythagorean theorem.

Sine Cosine Tangent

Sine, cosine, and tangent are trigonometric ratios that relate the angles and sides of a right triangle. These ratios are fundamental in understanding trigonometry.

• Sine (\(\sin\theta\)): This is the ratio of the length of the opposite side to the hypotenuse. In our example, \(\sin\theta = \frac{16}{20} = \frac{4}{5}\), indicating how steeply the ladder leans against the wall.
• Cosine (\(\cos\theta\)): This involves the ratio of the adjacent side to the hypotenuse. So, \(\cos\theta = \frac{12}{20} = \frac{3}{5}\) shows how much of the ladder lies along the ground.
• Tangent (\(\tan\theta\)): It's the ratio of the opposite side to the adjacent side. Here, \(\tan\theta = \frac{16}{12} = \frac{4}{3}\) illustrates the steepness of the angle between the ladder and the ground.

These ratios are used not only to find missing side lengths but also to determine angles within right triangles.

Pythagorean Theorem

The Pythagorean Theorem is a crucial formula in trigonometry for dealing with right triangles. It states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. The formula is:\[c^2 = a^2 + b^2\]The theorem was used in the original problem to determine the length of the ladder's bottom leg on the ground. By knowing the hypotenuse and one other side, we are able to calculate the unknown length conveniently:

• \( c = 20, a = 16 \)
• Using the theorem: \( b^2 = 20^2 - 16^2 = 400 - 256 = 144 \)
• Therefore, \( b = \sqrt{144} = 12 \)

This approach demonstrates how powerful the Pythagorean Theorem is for solving problems involving right triangles by helping find the missing side lengths easily.