Maths is both an interesting as well a scary subject. For those who find it easy, it becomes fun to learn new chapters. People who find it difficult to understand try to avoid solving maths questions as far as possible, especially if the problem belongs to trigonometry. Due to so many formulas and values of trigonometric angles, a lot of students face difficulty in solving trigonometry questions. It is hard to learn and grasp the concepts of trigonometry. We have come up with math exam tips from Cuemath that shall help you ace this topic.
Do you also get confused between cos and cot, sin, and sec? Or, do you not know how to begin with a trigonometry question? To help you get rid of this issue, we have curated a list of tips and tricks on how to solve trigonometry problems. Here is a step by step guide on how you can learn to solve trigonometry questions with ease.
- First, understand the importance of trigonometry- Students often face difficulty while solving trigonometry questions because they aren’t taught the use or benefit of this chapter. Once you get to know why you are learning the chapter, it becomes easier to focus on the solution. Trigonometry is a branch of mathematics that involves the calculation of angles of a right-angled triangle and the length of its sides. A right-angled triangle has two sides that are perpendicular to each other. It means that the two sides are standing at an angle of 90 degrees. Trigonometry is used to calculate the angles of the triangle.
- Understand what is hypotenuse, adjacent, and opposite- Once you are clear about why you are solving the question, the next step is to get your basics clear. Since trigonometry involves angles and lines, it is important to understand what is the hypotenuse, adjacent and opposite sides in a right-angled triangle. In the above point, you have already learned what is a right-angled triangle. The third side of a right-angled triangle is known as the hypotenuse. It is the longest side in a triangle. You may have come across this while studying the Pythagoras theorem.
Now to find out the adjacent and the opposite side in the triangle, you need to first mark the angle that is being asked. Since one angle is 90 degrees, the question will ask you to find the remaining two angles. In trigonometry, the angles of the triangle are marked as ‘theta’. Its symbol is θ. When you mark the theta in the triangle, the side forming the angle other than the hypotenuse is known as the adjacent side. The third remaining side is known as the opposite side. The adjacent side is also known as the base, whereas the opposite side is called the perpendicular.
- Remember the formulas- After learning to figure out the hypotenuse, adjacent and opposite sides, now it is the time to learn the formulas of trigonometry. These are simple formulas that you can learn easily. But, before moving on, you have to understand that there are six trigonometric ratios. Sin, cos, tan, and cosec, sec, cot. Cosec is the opposite of sin, sec is the opposite of cos, and cot is the opposite of tan. It means that:
- Cosec = 1/sin
- Sec = 1/cos
- Cot = 1/tan
Now, let us start with how you can calculate the values of these ratios.
- Sin θ = perpendicular side/hypotenuse divide by length of opposite side/hypotenuse
- Cos θ = base/hypotenuse divide by adjacent/hypotenuse
- Tan θ = perpendicular/base divide by opposite/adjacent side,
Now, cosec, sec, and cot will be the reverse of sin, cos, and respectively.
- Cosec θ = 1/sin = hypotenuse/opposite side
- Sec θ = 1/cos = hypotenuse/adjacent
- Cot θ = 1/tan = adjacent/opposite side
- Read the Questions Carefully– Now you know the formulas, and all your concepts are clear. You can now move ahead with solving the trigonometric questions. Make sure that you read each question at least twice to figure out what has been given and asked in the question. Sometimes, there might be extra values in the problem that are intentionally added to confuse the student. You should stick to the values or ratios that are required to solve the question. Also, at times you will come across questions where the lengths of the triangle will not be given directly in the question.
For example, there may be a question where the length of the adjacent and opposite sides will be given, and you will have to find out the length of the hypotenuse. You may also be asked to find out the value of sin θ or cos θ. In such a question, you will have to find out the hypotenuse by applying Pythagoras theorem.
Also, there are questions where an equilateral triangle or an isosceles triangle is given in the question. You will have to convert it into a right-angled triangle by dividing the triangle into two halves. You will then find out the lengths of the sides and the values of the angles.
- Draw a Diagram- After reading the question, the next step is to note down all the values that have been provided in the question, as well as what is to be found out, in your notebook. It gives you clarity and also ensures that you do not have to read the question again and again. It helps you solve the question without getting confused. If there is already a diagram or a triangle drawn in the book, you should copy that in your notebook. In the case where there is no diagram, you should draw one for your reference. Make a right-angled triangle, mark all the sides and angles. Then, write the values of the angles and the length of the sides if given in the question.
In case there is a question apart from the usual triangle problems, for example, where you have to find out the length of the shadow of a tree or where you have to find out the distance of a person who is from the ground or a pole, you will have to make a diagram. All you have to do is form a triangle in your mind and then draw it on the notebook.
- For proving questions, start from the side which is complex- Trigonometric questions are usually of two kinds. While the first one is discussed above, where you have to find the value of the angles, the next is proving questions where a trigonometric equation is given, and you have to prove that the left side is equal to the right side i.e., L.H.S = R.H.S.
Proving questions are usually more difficult to solve. To prevent getting stuck in the question, the trick is to first solve the side that is complex. For instance, you have a mix of cosec, sec, and tan on the right-hand side, whereas on the left side there is a simple equation involving sin and cos. In that case, you will first try to solve the complex side i.e., the right side. The logic behind this is that you should try to solve the complex equation using the formulas rather than making the easier side complex. You should not try converting the left side simple equation into a complex one, instead, start with the complex one so that you know what you have to derive from the equation.
- Convert the equation into sin and cos- Sin and Cos are the most commonly used trigonometric ratios. Most of the proving questions involve a sin and cos equation on either side. Therefore, you should try to convert the entire equation into sin and cos, so that it becomes easy to solve. If the questions involve Cosec or Sec, you should try to simplify it by converting it into sin and cos.
- Learn the values by heart- There is no such shortcut to solve trigonometric problems, especially the proving questions. You will have to learn the values by heart. It is extremely important in Inverse trigonometric questions, where you will have to learn a lot of formulas for solving particular questions. While applying the values or formulas in the solution, make sure that you are applying the correct value.
- Double-check your solution- This is the most important step. Since the names of trigonometric questions can be confusing, it is natural to make mistakes or mix up the names while solving the question. Therefore, it is important that you revise or double-check your questions once you are done with it. Check that you have applied the correct formula and values.
- Practise as many questions as you can– Lastly, you can only master trigonometric questions when you practise. There are never enough questions that you can practice. Solve as many of them as you can. While solving questions, make sure that you try all kinds of questions, and not focus on one or two. The more you solve, the more you will get better.
We hope that this helps you in solving trigonometry problems. If you have doubts related to this or any other queries, please type that in the comments below.