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Trigonometry is a specialised branch of mathematics that deals with the study of triangular sides and angles. At the same time, calculus is a branch of mathematics which keeps track of continuous change of values in variables. Calculus tries to use mathematics in solving problems raised in a dynamically changing environment, and trigonometry supports this by providing ease in huge calculations involved. These two fields are used to derive miraculous results when used together.

Trigonometry is misunderstood as a complex one while it simplifies the otherwise complex and repetitive tasks in calculus. It carried an important place in calculus because it simplifies the calculations, provides accurate and precise calculations. We would be reading more about its use and applications in specialised fields as well as daily life in the following article.

What is Trigonometry and How Does it Simplify Calculus?

Trigonometry is a study of triangular length, height, and angles. There are three sides of a triangle aligned at a different angle. All angles of the triangle should sum up to 180 degrees to qualify as a triangle. The length of the sides and angles decide on the type of triangle.

Trigonometry replaces normal variables used in calculus. It provides simplicity of using angles instead of sides or vice versa as per the parameters at hand. This can be best explained with the help of an equation of a unit circle. The unit circle is a circle with a radius of 1 unit. The equation of the unit circle can be written in terms of the point of intersection with the x-axis, and the point of intersection with the y-axis.

Equation of circle : x^2 + y^2 = 1

Here coordinates (x,y) are the points on the circumference of a circle which when connected with the centre of a circle forms a radius. As per the equation above, the sum of the square of x and y coordinates gives a radius of a circle. Now see another equation:

Equation of circle using trigonometric functions: Cos^2 (A) + Sin^2 (A) = 1.

Here angle “A” is an angle of radius drawn between the (x,y) coordinates and the centre of the circle. Even if we are not aware of (x,y) coordinates and know the angle, the trigonometric function can be used.

Similarly, there are various graphs associated with trigonometric ratios which can also be used during calculations in calculus rather than getting into finding infinitely small values to reach the approximation.

Excited for some hands-on experience of using trigonometry in calculus? Here is an example along with the explanation for you!

Example: Use calculus to balance the below equation in terms of trigonometric functions. The equation below is also an equation of derivatives in the standard calculus.

F`(x) = lim h-> 0 { ( f(x+h) – f(x) ) / h }

Solution:

Use sin (x) in the place of f (x) in RHS.

F`(x) = lim h-> 0 { ( sin(x+h) – sin(x) ) / h }

Appling the property of expansion of function sin (x+h)

F`(x) = lim h-> 0 { ( cos (x) . sin(h) + cos (h) . sin(x) – sin(x) ) / h }

Taking sin (x) common in the equation.

F`(x) = lim h-> 0 { ( cos (x) . sin(h) + (cos (h) – 1 )sin(x) ) / h }

Diving the whole equation into two parts with the limit of “h” tending to zero in both parts.

F`(x) = lim h-> 0 { ( cos (x) . sin(h) } + lim h-> 0 { (cos (h) – 1 )sin(x) ) / h }

F`(x) = cos(x) * 1 + sin(x) * 0

cos(x)

Use sin (x) in the place of f (x) in LHS. 

F`(sin (x) ) = cos (x).

RHS = LHS

Explanation: Let’s talk about the equation first. This equation explains the concept of continuous change which is captured in calculus. The LHS and RHS contained variables as a function of “x”. This represents a linear value that keeps changing as per the function defined. The calculations would have been far more complex if the values are used because of limitations in the use of values after the decimal point in a number. This shortcoming was eliminated with the use of trigonometric functions. The properties and rules of trigonometric functions simplify the calculations of such detail. The above example is easy to solve and understand as there are no complex calculations involved.

Real-Life Applications

Trigonometry has reduced the complications of calculus to produce some real-life applications:

    1. 1. Calculus is used to determine the impact of temperature, humidity, light, food source and the factors responsible for the rate growth of bacteria. These are extensive-time taking calculations involving various factors to be considered. Instead of using normal variables, trigonometric functions can be used as an abstraction layer. Trigonometric functions eliminate the need to calculate each small step, giving biologists time to concentrate on optimising the calculations.2. Imagine you are a part of a crime scene investigation where two cars collided. You are trying to establish a pattern out of the trajectory, speed, direction of the cars involved in such accidents. Calculus can be used here to calculate near to accurate values by aggregating the values obtained. These calculations can then be further optimised with the help of trigonometric functions.3. You all might have spent childhood playing video games, but now the overall presentation of video games has developed a lot.
      Multiple graphics and live effects are added to give you real life-like experience. This is possible with the use of calculus in
      designing responsive three-dimensional objects. These objects respond to all rapidly changing activities of a player. The response in such games is very precise by reaching the accurate directions, time, length, and other measurements involved which are derived with the help of trigonometry
    2. 4. Trigonometry and calculus are used in combination to calculate the distance of moving celestial bodies. The movement of celestial bodies relative to each other and earth is used by mathematical astronomers as well. Mathematical astronomers use the changes in planetary positions for predictions. One of Its uses can be found in the Copernican’s revolutionary model named “Sun-centered geometrical model”5. You might have come across multiple research papers and studies published by scholars about algae and photosynthesis in algae, but ever wondered how these marine engineers and scientists use calculus while studying algae? They capture changes for a long period of time and use differential and integrational calculus to derive insightful information. They use trigonometry in calculus to calculate the depth of sunlight in the ocean at different levels, which determines the rate of photosynthesis reaction in algae.6.  Calculus has been traditionally used in medical science. It is used to track the changes in a patient’s health over a period of time on the basis of different factors. Small changes in the nature of a virus can be tracked with the help of calculus. The pace of spread, impact on health, severity, and control measures can be determined using calculus coupled with trigonometry. Such changes in the biological composition of any bacteria or virus can cause havoc if not properly administered. Well, no doubts about this in the era of COVID-19!

      Some Useful Tips

      Here are some tips for you:

          1. Read through the problem statement thoroughly and determine what is provided while what is to be determined. For example, in some cases, angles are to be determined while in others, it is the length.
          2. Draw all possibilities in diagrams. Keep eliminating the diagrams which further do not meet your criteria.
          3. Visualise the rotating triangles in complex motion-related calculus questions. For example, suppose you rotate a triangle with one angle as the centre point. In that case, other angles lie on the circumference of a circle if the triangle is an “equilateral triangle”. Try this drawing on the piece of paper to get more insights. Isn’t it an interesting fact?
          4. Apply rules and laws wherever possible. Try to find out if a complex calculative equation can be reduced with the help of trigonometric ratios and laws.
          5. Rearrange equations to find the possibility of applying formulas.
          6. Always validate your answers with the help of geometrical diagrams. For example, you got 0 degrees as an answer, but is it practically possible to have zero degrees angle in a triangle?

      Conclusion

      Trigonometry is like a toolkit in calculus with the latest tools to solve your mathematical dilemma. It has an elegant way of solving complex calculations with the help of standardised rules. Trigonometric ratios are used to solve geometrical, calculative, measurement, and differentiation-related mathematical problems. The reason it becomes tough to digest for students is that they are not able to visualise triangles in trigonometry, and are unable to apply rules of trigonometry even when they know it.

      If you are ready to expand your horizons of visualisation and look for simple tools to solve complex calculus problems, then trigonometry is the right match for you. If you are looking for partners with whom you can integrate your success and differentiate your failures, then Cuemath is the perfect platform for you.