Why Do We Study Trigonometry – Practical Applications of Trigonometry – Part II

How Trigonometry Runs The Modern World

Mechanical and Motion

• The torque of the EV motor depends on the sine of the angle between the magnetic field and the rotor, so trigonometry is used every millisecond to keep this angle near the best value for maximum power and efficiency. Power electronics engineers use high-speed computer in EVs which monitors the position of the rotor thousands of times per second. It then shifts the electricity in the copper coils to ensure that the magnetic field is always leading the rotor by exactly 90°. By keeping the sine of the angle at 1, the EV gets that famous instant torque and smooth acceleration.
• Even though a combustion engine looks like a mechanical device, it is actually controlled by the mathematics of angles and waves. Every movement of the piston is managed by converting circular motion of the crankshaft into linear motion of the piston using sine and cosine functions.  For maximum power, the spark plug must fire a few degrees before the piston reaches the very top. Engineers determine rate of change (through Calculus) of these trigonometric functions to determine the velocity of the piston. Trigonometry in Engine Control Unit decides where the piston is, when the fuel should ignite, when the valves should open, and how smoothly the whole engine runs.
• When a robotic arm moves to pick up an object, each joint rotates by a certain angle. To know where the hand will end up, the robot uses sine and cosine to convert those joint angles into exact positions in space. This is how factory robots can weld, paint, or assemble things with millimetre accuracy.
• Trigonometry is also used when robots walk or roll. A walking robot must decide how much to lift and swing each leg so it does not fall. A wheeled robot must turn at the right angle to reach a target. The robot’s computer uses trigonometry to calculate how far it will move when the wheels rotate by a certain angle and how much it must turn to face a new direction. Robots also use trigonometry to see and sense. Cameras, lasers and ultrasonic sensors measure the angles and distances to objects. By using sine, cosine and tangent, the robot converts these angles into the exact location of walls, obstacles, and people. This allows self-driving robots, warehouse robots and delivery robots to map rooms, avoid collisions and move safely.
• A crane lifts a load using a long arm and a cable. The long arm is raised and lowered by changing its angle with respect to the ground. Using trigonometry, the crane’s onboard computer calculates how high the load will go and how far out from the crane it will be for any given arm length. For example, by using sine and cosine, the crane’s computer can find the vertical height of the load and its horizontal reach which helps the operator place materials exactly where they are needed on a building site. Modern cranes with computers and sensors can lift extremely heavy objects and still place them gently and accurately. They are continuously using trigonometry to turn angles into distances, forces, and safe movements.

• When a civil engineer designs a mountain road, they use tan(θ) to calculate the gradient. The tangent function is the most used non-wave function in engineering. It represents the slope or steepness. 
• In Computer Graphics, to make a 3D character look realistic, the computer needs to know how light hits a surface. It uses the tangent of the angle to calculate surface normals which determines where shadows should fall.

Navigation and Control

• In GPS navigation, trigonometry is what allows your phone or car to know your exact location and how to guide you to your destination. GPS satellites orbit high above the Earth and continuously send radio signals. Our phone receives these signals and measures how long each signal took to arrive. From this, it calculates the distance to each satellite. But to find our exact position, it also needs to know the angles at which those satellites are seen in the sky. Using trigonometry, the GPS system forms triangles between your phone and several satellites and solves these triangles using sine and cosine. The point where these distances and angles match is our precise location.

• As we move, GPS uses trigonometry again to calculate our direction and speed. By comparing our old and new positions, it finds the angle of our movement and determines whether we are going straight, turning left, or turning right. This is how the navigation app knows which road we are on and how far we are from the next turn. Even features like showing the shortest route or estimating arrival time depend on these angle-based calculations. 
• In avionics, trigonometry in flight control systems is what allows an aircraft to stay balanced, turn correctly and follow its planned path through the sky. Sensors inside the plane constantly measure angles such as how much the aircraft is tilted up or down, left or right. If a pilot tilts the plane too sharply without increasing total power or lift, a part of the lift gets diverted sideways. The plane thus starts to sink because the vertical component of the lifting force is no longer enough to fight gravity. The autopilot uses trigonometric calculations to automatically pull the nose up during a turn to maintain altitude.

• When a pilot is landing, they need to stay on a specific glide path. If the altitude is fixed at 1,000 feet, the cotangent tells them exactly how far from the runway they are horizontally. Cotangent function is used when the height is the fixed constant, and we are trying to find the base or the diagonal.
• In antenna design, radar antennas (like those at airports) are shaped using a cosecant-squared curve. This ensures that a plane flying at a constant altitude receives the same signal strength whether it is far away or directly overhead.
• In self driving cars, trigonometry is what allows the vehicle to understand where it is, where it is going, and how it should steer and move safely. The car uses cameras and radar to detect other vehicles, pedestrians, lanes, and obstacles. These sensors measure angles and distances to everything around the car. Using trigonometry, the car converts these angles into the exact positions of objects on the road. For example, it precisely calculates how far a pedestrian is, how wide a lane is and at what angle another car is approaching. This helps the computer build a map of the surroundings in real time.

Energy and Electricity

• In power grids, trigonometry is what allows electricity to be generated, transmitted, and delivered in a safe and efficient manner to millions of homes and industries. Electricity in power lines flows as AC waves which rise and fall in a sine-wave pattern. Using trigonometry, grid engineers describe the voltage and current at every moment using sine and cosine. Power stations, transformers and substations must all stay in perfect synchrony. Trigonometry is used to match the phase angles of different generators so they push power into the grid smoothly instead of colliding with each other. When we connect a generator to the national grid, we are actually merging two massive, vibrating energy systems. If their phase angles don’t match, the results are catastrophic. Trigonometry is also used to calculate real power, the electricity that does useful work and reactive power, the energy that is stored and returned by motors and transformers. These are found using cosine and sine of the phase angle between voltage and current. This is how trigonometry quietly runs every electrical appliance in the modern world.

• In solar power systems, trigonometry and calculus allow sunlight to be captured, converted and delivered as useful electrical energy. The amount of sunlight a solar panel receives depends on the angle at which sunlight hits it. If the panel is tilted at the wrong angle, much of the light is reflected away. Using trigonometry, controllers calculate the best tilt and direction of the panel so that the sunlight falls as directly as possible on its surface. The more directly the light hits, the more electricity the panel produces.

Communication

• Every radio signal is a sine wave and sine waves are pure trigonometry. Our voice is not a sine wave but a changing pattern. To send it, it is to be attached to sine wave. Modulation is the process of attaching a low frequency information signal (like voice) onto a high frequency carrier wave so that it can travel long distances through the air. In Amplitude Modulation (AM), the height of the sine wave is changed. Here trigonometry controls how tall the sine wave is at every moment. In Frequency Modulation (FM), the speed of the sine wave is changed. This is done by changing the phase angle of the sine wave. Since phase is an angle, trigonometry is what measures and controls this shift. The radio receiver compares the incoming signal with reference sine and cosine waves. From these calculations, it reconstructs the original sound. Sound is stored as changes in sine waves and receivers decode sine waves.
• Inside mobile phones and the cell tower, the signal is written as a combination of cosine and sine waves. By adjusting amplitudes and angles of these waves, the system can encode letters, images and videos into the radio signal. At the receiver, trigonometry is again used to compare the incoming wave with reference sine and cosine waves and extract the original information. Similarly, trigonometry is used in  Wi-Fi routers and Bluetooth.

Medical

• An MRI machine sends powerful radio waves into the body while it is inside a strong magnetic field. These radio waves are described using sine and cosine. When the waves interact with atoms in the body, they produce tiny returning signals that are also sine waves. These signals have different strengths and phase angles depending on where they come from in the body.
• Using trigonometry, the MRI computer separates these returning waves into cosine and sine parts. By measuring the angle and amplitude of these waves, the system can calculate exactly where in the body the signal came from and how strong it is. This allows the machine to build a detailed map of tissues slice by slice.
• In CT Scans, when an X-ray beam passes through the body at an angle, the computer uses secant functions to calculate exactly how much tissue the beam has to penetrate, which is vital for getting a clear image.

Space

• When a space telescope looks at a star, it does not just see it but also measures the angle at which the light from that star enters the telescope. This angle is extremely small because the star is very far away. Tiny sensors inside the telescope detect whether the star’s image is slightly left, right, up, or down from the centre of the camera. Using trigonometry, the telescope’s computer tells how much the telescope must rotate in each direction to bring the star back to the centre. For example, if the star appears a small angle to the left, trigonometry calculates how far the telescope must turn so that its optical axis lines up exactly with the star.

• This process happens continuously. As the telescope moves or as the Earth rotates and stars shift, the angles change slightly. Trigonometry keeps converting these small angle changes into precise corrections, allowing the telescope to stay perfectly pointed at the target for hours or even days. That is how space telescopes can take sharp and steady images of extremely distant objects. Similarly, using trigonometry, weather satellites convert the angle at which light or radar signals from clouds and the ground reach its sensors, into exact distances and locations, telling us precisely where a storm or cloud system is.
• In Atmospheric Science, when a satellite looks straight down at Earth, it looks through a certain thickness of atmosphere. But if it looks at an angle, it has to peer through more air. That extra distance is calculated using the secant of the angle. The secant function is used whenever we are traveling diagonally through a thickness.
• In space missions, trigonometry is what allows spacecraft to travel across millions of kilometres and reach the correct planet, moon or space station.
• When a spacecraft is flying through space, it cannot follow roads or rails. It moves by pointing its engines at very precise angles. Using trigonometry, mission computers calculate the exact direction in which the spacecraft must fire its engines so that the push sends it toward the desired target. Even a tiny error in angle can make the spacecraft miss its destination by thousands of kilometres.

When a spacecraft is millions of kilometres away, scientists cannot see it directly. They can only communicate with it using radio signals. These signals travel at the speed of light. When Earth sends a signal to the spacecraft and the spacecraft sends one back, the time taken tells scientists the distance between Earth and the spacecraft. But distance alone is not enough as scientists must also know which direction the spacecraft is in. Large radio antennas on Earth receive the signal from the spacecraft at slightly different angles. By measuring these tiny angle differences, scientists can tell where in the sky the spacecraft is locate