The trigonometric functions are the trigonometric ratios of a triangle. The trigonometric functions sine, cosine, and tangent, are abbreviated as sin, cos, and tan. Let’s look at how specific ratios or functions are measured in a right-angled triangle.

In a right-angled triangle, the longest side is the hypotenuse, with the adjacent and reverse sides being similar and opposite.

Some trigonometry angles 0 degrees, 30 degrees, 45 degrees, 60 degrees, and 90 degrees are widely seen in trigonometry issues. These angles’ trigonometric proportions, like sin, cosine, and tangent, are straightforward to remember. To locate certain angles, one must first outline a right-angled triangle within which one significant acute angle corresponds to the trigonometric angle. Those angles will be described in relation to the ratio concerned.

For instance in right-angled triangle,

Sin = Perpendicular/Hypotenuse

i.e. θ = (P/H) = sin-1

In the same way,

θ = (Base/Hypotenuse) = cos-1

θ = (Base/Perpendicular) = tan-1

Complementary angles are the sequence of two angles whose sum equals 90 degrees in mathematics. The angles 30° and 60°, for example, are complementary since their sum is 90°. It is observed in complementary triangles.

If two angles, say X and Y, are complementary, they are considered to be complementary.

X + Y = 90 degrees

X is defined as the counterpart of Y in this situation and vice versa.

Since the right angle calculation is set in a right-angle triangle, the other two angles still shape the complement since the number of angles in a triangle is 180°.

There are many real-world applications of trigonometry.

Suppose we have the building’s height and the angle created when an object is viewed from the top of the building. In that case, we may use the tangent function to find the difference between the object and the building’s bottom. For example, the tan of the angle equals the combination of the building’s height and the distance.

Oceanography, seismology, meteorology, physical sciences, physics, acoustics, mapping, communications, among other areas, have used trigonometry. It may also be used to calculate the height of a peak, the duration of a large river, among other related tasks.

• Aviation Trigonometry: In the last few years, aviation technology has advanced significantly. It took into account the wind’s pace, direction, distance, and the wind’s speed and direction. The wind has a significant influence on where and how a plane flies. Trigonometry can be used to solve this equation. For example, if an aircraft is travelling 250 miles per hour and the wind is moving due south at 19 miles per hour, it is 55 degrees north of east. This calculation will be solved using trigonometry to determine the third side of the triangle to guide the aircraft in the correct direction.
• Criminology Trigonometry: In the analysis of a crime scene, trigonometry is required. Trigonometry functions help calculate a projectile’s trajectory and estimate the causes of a collision in a car accident. It’s often used to determine if an obstacle falls or at what angle a gun is fired.
• Marine Biology Trigonometry: Marine biologists frequently use trigonometry to determine the depth of sunlight inhibiting algae’s ability to photosynthesis. Sea biologists measure the scale of bigger species like whales and consider their behaviour using the trigonometric function and mathematical models.
• Navigation Trigonometry: For navigating directions, trigonometry is used to estimate where to place the compass to get a straight line. It would be quick to pinpoint a position and calculate distance and see the horizon using a compass and trigonometric navigation functions.

In this chapter, we talked about trigonometry and its concepts in detail. We can solve simple 2D problems involving one right-angled triangle in the curriculum and apply them to real-life concepts with this knowledge.

1. What is the concept of trigonometry?
   Trigonometry is a subset of mathematics that studies the connection between the angles and the sides of a triangle (right triangle). The relationship between sides and angles is established for six trigonometric functions.
2. What is trigonometry’s primary purpose?
   The sine, cosine, and tangent functions are the three main functions in trigonometry.
3. What is the name of the individual who developed trigonometry?
   Hipparchus was a Greek astronomer, geographer, and mathematician who invented trigonometry.
4. What are the real-life applications of trigonometry?
   The estimation of height and distance is one of the most significant real-life uses of trigonometry. Aviation departments, navigation, criminology, and aquatic biology are a few of the fields utilising trigonometry.
5. What is the purpose of trigonometry?
   Students with solid trigonometry abilities can calculate complicated angles and proportions in a short amount of time. Trigonometry is a subset of mathematics widely utilised in construction, engineering, and many other sciences.