Basic Trigonometric Concepts in Applications
To navigate through the applications, it's pivotal to have a firm grasp on the basic trigonometric concepts. The primary trigonometric ratios, which will be our tools in solving application problems, are:
- Sine (sin): Opposite/Hypotenuse
- Cosine (cos): Adjacent/Hypotenuse
- Tangent (tan): Opposite/Adjacent
These ratios are instrumental in various height and distance problems, especially those involving right-angled triangles.
Application in Height Problems
Trigonometry is a key player in determining the heights of various objects without physically measuring them. This includes finding the height of a building, a tree, a mountain, etc., by forming right-angled triangles and using trigonometric ratios.
Example 1: Finding the Height of a Tree
Consider a scenario where you are standing at a point on the ground and looking up at the top of a tree with an angle of elevation of 60 degrees. If you are standing 30 meters away from the tree, you can find the height (h) of the tree using the tangent function:
tan(theta) = Opposite/Adjacent
tan(60 degrees) = h/30
h = 30 * tan(60 degrees)
Calculating the above, we find that tan(60 degrees) is equal to sqrt(3) or approximately 1.732. Therefore,
h = 30 * 1.732
h = 51.96 meters
Example 2: Finding the Height of a Building
Similarly, if a person observes the top of a building with an angle of elevation of 45 degrees and is standing 20 meters away from it, the height (H) of the building can be found using:
H = 20 * tan(45 degrees)
Since tan(45 degrees) = 1,
H = 20 meters
Application in Distance Problems
Trigonometry also assists in finding distances between objects, such as the distance of a ship from a lighthouse, the distance of a plane flying at a certain altitude, etc.
Example 3: Finding the Distance of a Ship from the Shore
If a lighthouse keeper observes a ship at sea with an angle of depression of 15 degrees and the lighthouse is 150 meters tall, the distance (D) of the ship from the base of the lighthouse can be found using:
tan(theta) = Opposite/Adjacent
tan(15 degrees) = 150/D
D = 150/tan(15 degrees)
Example 4: Finding the Distance between Two Planes
If two planes are flying at the same altitude and the angle formed at a point on the ground between them is 75 degrees, and the height at which they are flying is 8000 meters, the distance (d) between the two planes can be found using the law of sines or using the tangent function, depending on the given information.
Real-World Applications
Trigonometry is not just limited to academic problems but is widely used in various real-world scenarios:
- Astronomy: Astronomers use trigonometry to find the distance of stars and planets from the Earth.
- Navigation: Pilots and ship captains use trigonometry to navigate their path and to find the distance between their starting point and destination.
- Architecture: Architects use trigonometry to calculate structural load, roof slopes, ground surfaces, etc.
- Engineering: Engineers use trigonometry to analyze forces, to design structures, and in various mechanical applications.
Practice Problems
Problem 1
A person looks at the top of a tower with an angle of elevation of 30 degrees. If the person is standing 50 meters away from the tower, find the height of the tower.
Solution:
To find the height of the tower, we can use the tangent function from trigonometry. The tangent of an angle is equal to the opposite side divided by the adjacent side in a right-angled triangle. Given an angle of elevation of 30 degrees and a distance of 50 meters from the tower, we can rearrange the formula to find the height (H) of the tower:
H = tan(30 degrees) x 50 meters
Calculating this:
H = 0.577 x 50 (since tan(30 degrees) is approximately 0.577)
H = 28.85 meters
So, the height of the tower is approximately 28.85 meters.
Problem 2
A pilot is flying over a straight highway. He determines the angles of depression to two mileposts, 6.6 km apart, to be 32 degrees and 48 degrees. Find the altitude of the plane.
Solution:
- Understand the Problem:
- Plane above a point (A) between two mileposts (B and C).
- Angles of depression to B and C are 32 and 48 degrees, respectively.
- Distance BC is 6.6 km.
- Define Variables:
- Altitude (h) is the height of the plane from point D on BC.
- x = BD and y = DC.
- alpha = 32 degrees, beta = 48 degrees.
- Use Trigonometry:
- tan(alpha) = h/x and tan(beta) = h/y.
- x + y = 6.6 km.
- Express x and y in Terms of h:
- x = h / tan(alpha) and y = h / tan(beta).
- Solve for h:
- Substitute x and y in x + y = 6.6 to get h / tan(alpha) + h / tan(beta) = 6.6.
- Using tan(32 degrees) ≈ 0.625 and tan(48 degrees) ≈ 1.11, we get:
- h / 0.625 + h / 1.11 = 6.6.
- Solve for h: h ≈ 2.64 km.
So, the altitude of the plane is approximately 2.64 km.
FAQ
The tangent ratio, defined as the ratio of the opposite side to the adjacent side in a right-angled triangle, is crucial in various real-world applications, such as in architecture and engineering. For example, engineers may use the tangent ratio to determine the slope of a ramp or a roof by comparing the vertical rise to the horizontal run. Similarly, in physics, the tangent of the angle of inclination provides valuable information about the velocity and trajectory of projectiles. Thus, understanding and applying the tangent ratio is vital in constructing structures and predicting physical phenomena accurately.
Trigonometric ratios are intrinsically linked to the unit circle, which is a circle with a radius of 1 unit, centred at the origin of a coordinate plane. The sine and cosine values of an angle in standard position (vertex at the origin and initial side along the positive x-axis) are defined as the y and x coordinates, respectively, of the point where the terminal side of the angle intersects the unit circle. The tangent value is the slope of the terminal side, which is the ratio of sine to cosine. Understanding these relationships provides a geometric perspective to trigonometric ratios and aids in visualising and solving problems involving periodic phenomena, waveforms, and circular motion.
The sine function is fundamental in modelling sound waves due to its periodic and oscillatory nature. Sound waves are essentially pressure waves that propagate through a medium, and they can be represented mathematically as sinusoidal waves. The sine function describes the displacement of particles in the medium as a function of time and space, where the amplitude represents the maximum displacement (loudness), and the period indicates the duration of one complete cycle (pitch). Thus, trigonometric functions, particularly the sine function, enable the analysis and synthesis of sound waves, contributing to advancements in acoustics, music technology, and audio engineering.
Co-functions in trigonometry are pairs of functions related by the equation cofunction(angle) = function(90 degrees - angle). Specifically, sine and cosine are co-functions, as are secant and cosecant, and tangent and cotangent. This relationship is derived from the complementary angles in a right-angled triangle, where the sum of two non-right angles is 90 degrees. Co-functions are utilised to simplify expressions and solve equations, especially when dealing with complementary angles. Understanding co-functions enables the transformation of trigonometric expressions into equivalent forms, providing alternative perspectives and methods to approach and solve trigonometric problems.
Trigonometric ratios, particularly sine, cosine, and tangent, are pivotal in navigation, especially in determining the direction and distance of travel. For instance, sailors and pilots utilise trigonometry to calculate their position and direction by measuring the angle between a distant object (like a star or a lighthouse) and a fixed baseline (often the horizon). By using these angles and some reference distances, they can triangulate their exact position on the sea or in the air, ensuring accurate navigation. This method is particularly crucial when GPS systems are unavailable or unreliable, providing an alternative, mathematical means of determining location and enabling precise travel.
Practice Questions
The surveyor forms a right-angled triangle with the building and the line of sight to the top of the building. To find the height (h) of the building, we can use the tangent trigonometric ratio, which is defined as tan(theta) = Opposite/Adjacent. Here, the opposite side is the height of the building, and the adjacent side is the distance from the surveyor to the building.
tan(75 degrees) = h/60
To find the height (h), we rearrange the formula:
h = 60 * tan(75 degrees)
Calculating the above, we find that tan(75 degrees) is approximately 3.732. Therefore,
h = 60 * 3.732
h = 223.92 meters
Let's denote the height from which the pilot observes the airports as h, and the distances from the plane to airports A and B as a and b respectively. We can use the tangent trigonometric ratio, defined as tan(theta) = Opposite/Adjacent, to set up the equations:
tan(12 degrees) = h/atan(7 degrees) = h/b
We also know that the distance on the ground between A and B is 150 km, so we can say that:
a + b = 150
We have three equations based on the problem statement:
- tan(12 degrees) = h/a
- tan(7 degrees) = h/b
- a + b = 150 km
We can rearrange equations 1 and 2 to express 'a' and 'b' in terms of 'h': a = h / tan(12 degrees) b = h / tan(7 degrees)
Now, substituting these into equation 3, we get: h / tan(12 degrees) + h / tan(7 degrees) = 150
To find the value of 'h', we can rearrange the equation as follows: h = 150 / (1/tan(12 degrees) + 1/tan(7 degrees))
Calculating the numerical value, we found that: h ≈ 11.67 km
So, the altitude at which the plane is flying is approximately 11.67 kilometers.
