Complementary Angles, Heights and Distances, Applications of Trigonometry

Complementary Angles

In trigonometry, angles play a crucial role. Two angles are considered complementary if their sum equals 90 degrees. For instance, if angle A and angle B are complementary, then A + B = 90°.

This relationship is fundamental because it leads to some important trigonometric identities. Let's consider a right-angled triangle ABC, where angle B is 90 degrees. Let angle C be denoted by θ. Then, angle A will be 90° - θ, as the sum of angles in a triangle is 180° (A + B + C = 180°, so A + 90° + θ = 180°, which means A = 90° - θ).

In this right-angled triangle:

• The side opposite to angle θ (angle C) is AB.
• The side adjacent to angle θ (angle C) is BC.
• The hypotenuse is AC.

Now, let's consider the trigonometric ratios for angle θ (angle C):

• sin θ = Opposite / Hypotenuse = AB / AC
• cos θ = Adjacent / Hypotenuse = BC / AC
• tan θ = Opposite / Adjacent = AB / BC

Next, let's look at the trigonometric ratios for angle (90° - θ) (angle A):

• The side opposite to angle (90° - θ) is BC.
• The side adjacent to angle (90° - θ) is AB.
• The hypotenuse is still AC.

So, for angle A (90° - θ):

• sin (90° - θ) = Opposite / Hypotenuse = BC / AC
• cos (90° - θ) = Adjacent / Hypotenuse = AB / AC
• tan (90° - θ) = Opposite / Adjacent = BC / AB

By comparing these ratios, we can derive the following identities:

• sin θ = BC / AC and cos (90° - θ) = AB / AC. This doesn't directly match. Let's re-evaluate the sides.

Let's be very clear about the sides relative to each angle:

For angle C (θ): Opposite = AB, Adjacent = BC, Hypotenuse = AC

For angle A (90° - θ): Opposite = BC, Adjacent = AB, Hypotenuse = AC

Now, let's re-compare:

• sin θ = AB / AC
• cos (90° - θ) = AB / AC

Therefore, sin θ = cos (90° - θ).



• cos θ = BC / AC
• sin (90° - θ) = BC / AC

Therefore, cos θ = sin (90° - θ).



• tan θ = AB / BC
• cot (90° - θ) = Opposite of (90-θ) / Adjacent of (90-θ) = BC / AB. Oh, wait. The cotangent of (90-θ) is Adjacent of (90-θ) / Opposite of (90-θ) = AB / BC.

Therefore, tan θ = cot (90° - θ).



• cot θ = BC / AB
• tan (90° - θ) = Opposite of (90-θ) / Adjacent of (90-θ) = BC / AB.

Therefore, cot θ = tan (90° - θ).



• sec θ = AC / BC
• csc (90° - θ) = Hypotenuse / Opposite of (90-θ) = AC / BC.

Therefore, sec θ = csc (90° - θ).



• csc θ = AC / AB
• sec (90° - θ) = Hypotenuse / Adjacent of (90-θ) = AC / AB.

Therefore, csc θ = sec (90° - θ).

These are the co-function identities, which are extremely useful in simplifying trigonometric expressions and solving problems, especially in heights and distances.

Heights and Distances

The concepts of trigonometry, especially the relationships involving angles and sides of right-angled triangles, are directly applied to solve problems involving heights and distances. We often encounter situations where we need to find the height of a tall object (like a tower, tree, or building) or the distance between two points, but direct measurement is difficult or impossible. In such cases, we use angles of elevation and depression.

Angle of Elevation

Imagine you are standing on the ground and looking up at the top of a building. The angle formed between the horizontal line of sight from your eye to the base of the building and the line of sight from your eye to the top of the building is called the angle of elevation. It's the angle measured upwards from the horizontal.

Conditions for Angle of Elevation:

• The observer is below the object being observed.
• The angle is measured upwards from the horizontal line of sight.

Example: If you are 100 meters away from the base of a tower and you look up at the top of the tower, the angle your line of sight makes with the horizontal ground is the angle of elevation.

Angle of Depression

Now, imagine you are at the top of a tall building or a cliff, and you look down at an object on the ground. The angle formed between the horizontal line of sight from your eye and the line of sight from your eye down to the object is called the angle of depression. It's the angle measured downwards from the horizontal.

Conditions for Angle of Depression:

• The observer is above the object being observed.
• The angle is measured downwards from the horizontal line of sight.

Example: If you are on a lighthouse 50 meters above sea level and you see a boat at sea, the angle you look down from the horizontal towards the boat is the angle of depression.

Important Note: The horizontal line of sight for the observer looking up (angle of elevation) is parallel to the horizontal line of sight for the observer looking down (angle of depression). Because these lines are parallel, the angle of elevation from the ground to an object is equal to the angle of depression from the object to the ground observer. This is due to the property of alternate interior angles being equal when a transversal line (the line of sight) intersects two parallel lines (the horizontal lines).

Example 1: Height of a Tower

A person stands 50 meters away from the base of a tower. The angle of elevation to the top of the tower is 60°. Find the height of the tower.

• Diagram: Draw a right-angled triangle. Let the height of the tower be 'h' (opposite side). The distance from the person to the base of the tower is 50 meters (adjacent side). The angle of elevation is 60°.
• Trigonometric Ratio: We have the adjacent side and need to find the opposite side. So, we use the tangent ratio.
• Equation: tan 60° = Opposite / Adjacent = h / 50
• Solve: We know tan 60° = √3. So, √3 = h / 50. Therefore, h = 50√3 meters.

Example 2: Distance from a Cliff

From the top of a cliff 100 meters high, the angle of depression to a boat is 30°. Find the distance of the boat from the base of the cliff.

• Diagram: Draw a right-angled triangle. The height of the cliff is 100 meters (one vertical side). Let the distance of the boat from the base of the cliff be 'd' (horizontal side). The angle of depression is 30°. The horizontal line from the top of the cliff is parallel to the sea level. The angle of elevation from the boat to the top of the cliff will also be 30° (alternate interior angles).
• Trigonometric Ratio: We have the side opposite to the 30° angle (height of the cliff) and need to find the adjacent side (distance 'd'). So, we use the tangent ratio.
• Equation: tan 30° = Opposite / Adjacent = 100 / d
• Solve: We know tan 30° = 1/√3. So, 1/√3 = 100 / d. Therefore, d = 100√3 meters.

Applications of Trigonometry

Trigonometry is not just about triangles and angles; it has wide-ranging applications in various fields:

1. Navigation and Surveying

Trigonometry is essential for determining distances and positions. Sailors and pilots use it to navigate, calculating bearings and distances. Surveyors use it to measure land boundaries, plot maps, and determine the heights of inaccessible points.

Example: A surveyor might measure the angle of elevation to the top of a mountain from two different points on the ground. Using trigonometry, they can calculate the height of the mountain and its distance from these points.

2. Astronomy

Ancient astronomers used trigonometry to calculate the distances to stars and planets, the sizes of celestial bodies, and their movements. Even today, it plays a role in understanding celestial mechanics and designing telescopes.

Example: Measuring the parallax angle of a star (the apparent shift in its position as the Earth orbits the Sun) allows astronomers to calculate its distance using trigonometry.

3. Physics and Engineering

Trigonometry is used extensively in physics to analyze forces, waves (sound, light, water), oscillations, and projectile motion. Engineers use it in designing structures, bridges, and machines, ensuring stability and calculating loads.

Example: When analyzing the forces acting on a bridge support, trigonometry helps resolve the forces into horizontal and vertical components.

4. Computer Graphics and Game Development

In creating 2D and 3D graphics, trigonometry is used for rotations, transformations, and calculating positions of objects on a screen.

Example: To rotate an object in a video game, trigonometric functions are used to calculate the new coordinates of its vertices.

5. Music and Sound Waves

The study of sound waves often involves trigonometric functions (like sine waves) to represent their amplitude, frequency, and phase.

6. Cartography (Map Making)

Creating accurate maps of the Earth's surface relies heavily on trigonometric principles to project a curved surface onto a flat plane.

Specific Trigonometric Identities for Heights and Distances

While the basic trigonometric ratios are sufficient, sometimes specific identities can simplify calculations:

• tan θ = cot (90° - θ): This is particularly useful when angles are complementary. For example, if you have an angle of 30° and another of 60° in a problem, you can use this identity.
• h = d * tan θ: Where 'h' is the height and 'd' is the horizontal distance. This is the direct application of the tangent ratio in heights and distances.
• h = (d₁ + d₂) * tan θ: If an object's height is observed from two different distances, d₁ and d₂, from the base, and the angles of elevation are θ₁ and θ₂.

Example 3: Using Complementary Angles

The angle of elevation of the top of a tower from two points P and Q at distances 'a' and 'b' respectively from the base and in the same straight line with it are complementary. Prove that the height of the tower is √(ab).

• Let the height of the tower be 'h'. Let the angles of elevation from P and Q be α and β.
• Given that P and Q are at distances 'a' and 'b' from the base.
• Given that α and β are complementary, so α + β = 90°.
• From point P: tan α = h / a
• From point Q: tan β = h / b
• Since α and β are complementary, we know that tan α = cot β.
• Substituting the expressions for tan α and tan β: h / a = cot β.
• We also know that cot β = 1 / tan β. So, h / a = 1 / (h / b) = b / h.
• Therefore, h / a = b / h.
• Cross-multiplying gives: h * h = a * b => h² = ab.
• Taking the square root of both sides: h = √(ab).

This proves the statement.

Example 4: Angle of Depression from a Building

A flagstaff stands on the top of a 30-meter high building. From a point on the ground, the angle of elevation of the bottom of the flagstaff is 30° and that of the top of the flagstaff is 60°. Find the height of the flagstaff.

• Let the height of the building be AB = 30 m. Let the flagstaff be BC, with height 'h'. The total height AC = AB + BC = 30 + h.
• Let the point on the ground be D. The distance of this point from the base of the building is BD = x.
• Angle of elevation of the bottom of the flagstaff (point B) is ∠ADB = 30°.
• Angle of elevation of the top of the flagstaff (point C) is ∠ADC = 60°.
• From triangle ABD: tan 30° = AB / BD = 30 / x.
• We know tan 30° = 1/√3. So, 1/√3 = 30 / x => x = 30√3 meters.
• From triangle ACD: tan 60° = AC / BD = (30 + h) / x.
• We know tan 60° = √3. So, √3 = (30 + h) / x.
• Substitute the value of x: √3 = (30 + h) / (30√3).
• Multiply both sides by 30√3: √3 * 30√3 = 30 + h.
• 90 = 30 + h.
• Solving for h: h = 90 - 30 = 60 meters.

The height of the flagstaff is 60 meters.