Tree Shadow Height And Trigonometry Problem Solving

To calculate the height of the tree, we can use the principles of similar triangles and the Pythagorean theorem. The scenario described forms two right-angled triangles: one large triangle formed by the tree, its shadow, and the rope, and a smaller triangle formed by John's height and the corresponding segment of the rope and shadow. By comparing these triangles and applying relevant mathematical concepts, we can determine the unknown height of the tree. This problem not only tests understanding of geometry but also the ability to apply these concepts in practical scenarios.

Understanding the Geometry

Let's break down the problem step by step. The tree, its shadow, and the rope form a right-angled triangle. Similarly, John's height, the segment of the shadow up to where he stands, and the rope segment up to his hand also form a right-angled triangle. These two triangles are similar because they share the same angle at the top (where the sun's rays hit) and both have a right angle. This similarity is crucial because it allows us to set up proportions between corresponding sides of the triangles.

The larger triangle has a base (shadow) of 20 meters and a hypotenuse (rope) of 25 meters. The height of this triangle is the height of the tree, which we need to find. The smaller triangle has a height of 1.8 meters (John's hand height). We need to find the length of the shadow segment corresponding to John's height to use the similarity of triangles effectively. We can also consider using the Pythagorean theorem for the larger triangle to find the tree's height directly if we knew the base and the hypotenuse, which we do. The strategy involves first using the Pythagorean theorem and then comparing the ratios of corresponding sides in the similar triangles to cross-verify or adjust our results.

Applying the Pythagorean Theorem

The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Mathematically, this is represented as: $a^2 + b^2 = c^2$, where $c$ is the hypotenuse, and $a$ and $b$ are the other two sides. In our case, the shadow (20 meters) and the height of the tree are the two sides, and the rope (25 meters) is the hypotenuse.

Let's denote the height of the tree as $h$. Applying the Pythagorean theorem to the larger triangle, we get:

$20^2 + h^2 = 25^2$

$400 + h^2 = 625$

$h^2 = 625 - 400$

$h^2 = 225$

Taking the square root of both sides, we find:

$h = \sqrt{225}$

$h = 15$ meters

So, according to the Pythagorean theorem, the height of the tree is 15 meters. This calculation gives us a direct answer based on the dimensions provided for the larger triangle.

Using Similar Triangles

Now, let's consider the smaller triangle formed by John's height and the end of the shadow. We know John's hand is 1.8 meters above the ground. To use the properties of similar triangles, we need to relate the ratios of corresponding sides in both triangles. Let's denote the length of the shadow segment corresponding to John's height as $x$. The ratio of John's height to the tree's height should be equal to the ratio of the shadow segment $x$ to the total shadow length (20 meters).

First, we need to find the distance from the base of the tree to where John is standing. This requires a bit more thought, as we haven't directly calculated this distance yet. The similar triangles approach would typically involve setting up the proportion:

$\frac{\text{John's height}}{\text{Tree's height}} = \frac{\text{Shadow segment corresponding to John}}{\text{Total shadow length}}$

However, to effectively use this proportion, we need to know the shadow segment corresponding to John. Without additional information or a clearer geometric relationship involving John's position relative to the shadow, this approach becomes less straightforward. The Pythagorean theorem directly gives us the height of the tree using the given dimensions of the larger triangle, making it the more effective method in this case.

Final Answer

Therefore, the height of the tree, calculated using the Pythagorean theorem, is 15 meters. The approach using similar triangles is conceptually valid but requires additional steps or information that are not directly provided in the problem statement. The direct application of the Pythagorean theorem provides a clear and efficient solution for this problem.

Finding cos θ when tan θ is Known A Comprehensive Guide

In trigonometry, understanding the relationships between different trigonometric functions is crucial for solving problems efficiently. This question explores the relationship between tan θ and cos θ. Knowing the value of one trigonometric function often allows us to determine the values of others using trigonometric identities and the fundamental definitions of these functions within a right-angled triangle. This problem exemplifies a common type of question in trigonometry where we leverage given information to deduce related trigonometric values.

Understanding Trigonometric Ratios

Before diving into the solution, it's essential to understand the basic trigonometric ratios. In a right-angled triangle, these ratios relate the angles to the sides of the triangle. The three primary trigonometric functions are sine (sin), cosine (cos), and tangent (tan). They are defined as follows:

• sin θ = Opposite side / Hypotenuse
• cos θ = Adjacent side / Hypotenuse
• tan θ = Opposite side / Adjacent side

The mnemonic SOH-CAH-TOA is often used to remember these ratios:

• SOH: Sine = Opposite / Hypotenuse
• CAH: Cosine = Adjacent / Hypotenuse
• TOA: Tangent = Opposite / Adjacent

In this question, we are given tan θ = 5/12. This means that in a right-angled triangle, the ratio of the side opposite to angle θ to the side adjacent to angle θ is 5/12. Our goal is to find the value of cos θ, which is the ratio of the adjacent side to the hypotenuse. To find cos θ, we need to determine the length of the hypotenuse using the information given for tan θ.

Using the Pythagorean Theorem

To find the hypotenuse, we can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). The formula is: $a^2 + b^2 = c^2$

In our case, we can consider the opposite side as 5 units and the adjacent side as 12 units (since tan θ = Opposite/Adjacent = 5/12). Let's calculate the hypotenuse (h):

$5^2 + 12^2 = h^2$

$25 + 144 = h^2$

$169 = h^2$

Taking the square root of both sides, we get:

$h = \sqrt{169}$

$h = 13$

So, the hypotenuse of the triangle is 13 units. Now that we have the length of the hypotenuse, we can calculate cos θ.

Calculating cos θ

Now that we know the adjacent side (12 units) and the hypotenuse (13 units), we can find cos θ using its definition:

cos θ = Adjacent side / Hypotenuse

cos θ = 12 / 13

Therefore, the value of cos θ is 12/13. This result gives us the cosine of the angle θ based on the given tangent and the properties of right-angled triangles. Understanding these relationships is fundamental in trigonometry and allows us to solve a variety of problems efficiently.

Summarizing the Solution

To recap, we started with the given value of tan θ = 5/12. We recognized that tan θ represents the ratio of the opposite side to the adjacent side in a right-angled triangle. Using the Pythagorean theorem, we calculated the hypotenuse of the triangle. Once we had the hypotenuse, we used the definition of cos θ (Adjacent side / Hypotenuse) to find its value. This step-by-step approach demonstrates how trigonometric ratios and the Pythagorean theorem can be combined to solve trigonometric problems effectively. The final answer, cos θ = 12/13, provides a clear understanding of the relationship between tan θ and cos θ in this specific scenario.