Problem 21 - Algebra 2 and Trigonometry

Question

An engineer wants to determine \(C D\) , the exact height of a building. To do this, he first locates \(B\) on \(\overline{C D}\) , a point 30 feet above \(C\) at the foot of the building. Then he locates \(A,\) a point on the ground 40 feet from \(C\) . From \(A\) , the engineer then finds that the angle of elevation of \(D\) is \(45^{\circ}\) larger than \(\theta\) , the angle of elevation of \(B .\) a. Find \(A B, \sin \theta,\) and \(\cos \theta\) b. Use \(\sin \theta\) and \(\cos \theta\) found in a to find the exact value of \(\cos \left(\theta+45^{\circ}\right) .\) c. Use the value of \(\cos \left(\theta+45^{\circ}\right)\) found in \(\mathbf{b}\) to find \(A D .\) d. Find \(C D,\) the height of the building.

Step-by-Step Solution

Answer

The building height, CD, is approximately 242.84 feet.

Step 1: Define Known Values

We know from the problem that the point \( B \) is 30 feet above \( C \), meaning \( BC = 30 \) feet. The point \( A \) is located 40 feet from \( C \), so \( AC = 40 \) feet.

Step 2: Calculate AB

Use the Pythagorean theorem on triangle \(\triangle ABC\):\[AB = \sqrt{AC^2 + BC^2} = \sqrt{40^2 + 30^2} = \sqrt{1600 + 900} = \sqrt{2500} = 50 \, \text{feet}\]

Step 3: Determine \( \sin \theta \) and \( \cos \theta \)

From triangle \(\triangle ABC\), where \( \tan \theta = \frac{BC}{AC} \), we have:\[\tan \theta = \frac{30}{40} = \frac{3}{4}\]Using the identity \( \tan \theta = \frac{\sin \theta}{\cos \theta} \):\[\frac{\sin \theta}{\cos \theta} = \frac{3}{4} \, \Rightarrow \, 3 \cos \theta = 4 \sin \theta\]Solving for \( \sin \theta \) and \( \cos \theta \) using \( \sin^2 \theta + \cos^2 \theta = 1 \):Assume \( \cos \theta = x \), then \( \sin \theta = \frac{3}{4}x \), giving:\[\left(\frac{3}{4}x\right)^2 + x^2 = 1\]\[\frac{9}{16}x^2 + x^2 = 1\]\[\frac{25}{16}x^2 = 1\]\[x^2 = \frac{16}{25}\]\[x = \frac{4}{5}\]Thus, \( \cos \theta = \frac{4}{5} \) and \( \sin \theta = \frac{3}{5} \).

Step 4: Calculate \( \cos(\theta + 45^\circ) \)

Using the identity for the cosine of a sum \( \cos(\theta + 45^\circ) = \cos\theta \cos 45^\circ - \sin\theta \sin 45^\circ \):\[\cos(\theta + 45^\circ) = \left(\frac{4}{5}\right)\left(\frac{\sqrt{2}}{2}\right) - \left(\frac{3}{5}\right)\left(\frac{\sqrt{2}}{2}\right)\]\[\cos(\theta + 45^\circ) = \frac{4\sqrt{2}}{10} - \frac{3\sqrt{2}}{10} = \frac{\sqrt{2}}{10}\]

Step 5: Find AD

Use the cosine value of angle \( (\theta+45^\circ) \) in \( \triangle ACD \) where\[\cos(\theta + 45^\circ) = \frac{AC}{AD} = \frac{40}{AD}\]Solve for \( AD \):\[\frac{\sqrt{2}}{10} = \frac{40}{AD} \, \Rightarrow \, AD = \frac{40 \times 10}{\sqrt{2}} = \frac{400}{\sqrt{2}}\]Rationalize the denominator:\[AD = \frac{400}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = 200\sqrt{2}\]

Step 6: Find CD, the Height of the Building

Since \( CD = AD - AC \):\[CD = 200\sqrt{2} - 40\]Calculate \( CD \):Since \( 200\sqrt{2} \approx 282.84 \) (using \( \sqrt{2} \approx 1.414 \)),\( CD \approx 282.84 - 40 = 242.84 \) feet.

Key Concepts

Angle of Elevation

The angle of elevation is a crucial concept in trigonometry, especially when determining heights and distances. It refers to the angle between the horizontal line of sight and the line of sight up to an object. Essentially, when you stand at point A and look up at the top of a building, the angle formed between your gaze and the ground is the angle of elevation.

It's important to measure this angle when you're interested in how high something is from your position on the ground. In our exercise, the engineer stands a certain distance away from the building and looks up to measure the angle of elevation.

• Point A is on the ground.
• Point B and D are above point C, the foot of the building.

The engineer then uses this angle to help determine other measurements and ultimately find the building's height.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry used to relate the sides of a right triangle. The theorem is stated as: \[ a^2 + b^2 = c^2 \] Where \( a \) and \( b \) are the legs of a right triangle, and \( c \) is the hypotenuse.

In our problem, points A, B, and C form a right triangle, with A and C on the ground and B being elevated at the height of 30 feet above point C. The distance between A and C is 40 feet, making it one of the legs, while BC, the height from C to point B, is the second leg.

By using the Pythagorean theorem, we find the length of the hypotenuse, AB, which helps us determine other trigonometric values like \( \sin \theta \) and \( \cos \theta \). This calculation is essential for further trigonometric analysis, such as using trigonometric identities to find the cosine of another angle.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables within their domains. These identities simplify the calculation of angles and are extremely useful.

For instance, the identity \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) helps us express one trigonometric function in terms of another. In our exercise, after determining \( \tan \theta \) as the ratio of the legs of the triangle, we use this identity to find \( \sin \theta \) and \( \cos \theta \).

Additionally, the Pythagorean identity, \( \sin^2 \theta + \cos^2 \theta = 1 \), is used here to conclude the exact values of \( \sin \theta \) and \( \cos \theta \), once we assume a variable value for either cosine or sine. This underscores how identities allow us to manipulate and solve complex trigonometric problems more smoothly.

Cosine of a Sum Formula

The cosine of a sum formula is an identity that expresses the cosine of the sum of two angles in terms of the sines and cosines of the individual angles. It is given by: \[ \cos(a + b) = \cos a \cdot \cos b - \sin a \cdot \sin b \] This formula becomes instrumental in calculating the cosine of an angle that is a sum of two known angles.

In this exercise, we're asked to find \( \cos(\theta + 45^\circ) \). By using our already calculated values of \( \sin \theta \) and \( \cos \theta \), along with known trigonometric values for \( 45^\circ \), we can straightforwardly find \( \cos(\theta + 45^\circ) \).

Understanding and applying this formula is crucial in many physics and engineering applications, including determining direction and positioning for construction, as seen with our engineer's task.