Introduction to Trigonometry – Formulas and Examples

Trigonometry is a branch of mathematics concerned with angle measurement and problems involving related angles. The Greek terms trigonon and metron, which refer to a triangle and measure, respectively are merged to form the word trigonometry. As a result, trigonometry refers to the study of relationships between a triangle’s sides and angles. So, in this Trigonometry article, we will learn about the Introduction to Trigonometry, trigonometric ratios, trigonometric ratios of some specific angles, trigonometric ratios related to complementary angles and trigonometric identities, along with the solved examples.

What is Trigonometry?

Assume that a group of pupils from a school is visiting Qutub Minar. Now, if a learner looks up at the top of the Minar, they can visualise forming a right triangle, as seen in the figure. Is it possible for the learner to determine the height of the Minar without having to measure it?

In the case described above, the distances or heights can be calculated using mathematical techniques from the field of mathematics known as trigonometry.

Trigonometric Ratios

The ratio of any two sides of a right-angled triangle can be used to derive trigonometric ratios. The Pythagoras theorem can be used to calculate the length of the third side of a triangle if the lengths of the first two sides are known.

We use the abbreviated version of the trigonometric ratios. The angle \(\theta\) is an acute angle (\(\theta<90^{\circ}\)) and is measured about the positive \(X\)-axis, in the anticlockwise direction.

Let us take a right-angled triangle \(\triangle A B C\) as shown.

Here, \(\angle C A B\) or \(\angle A\) is an acute angle. The side \(B C\) is facing \(\angle A\). So, we call it the side opposite to \(\angle A\) or perpendicular. \(A C\) is the hypotenuse of the triangle \(\triangle A B C\), and the side \(A B\) is a part of \(\angle A\). So, we call it the side adjacent to \(\angle A\) or the base.

The trigonometric ratios of the angle \(\theta\) in the right triangle \(A B C\) are defined as follows:
(i) \(\sin \theta=\frac{\text { Side opposite to } \angle A}{\text { Hypotenuse }}=\frac{ \text { Perpendicular }}{\text { Hypotenuse }}=\frac{B C}{A C}\)
(ii) \(\cos \theta=\frac{\text { Side adjacent to } \angle A}{\text { Hypotenuse }}=\frac{\text { Base }}{\text { Hypotenuse }}=\frac{A B}{A C}\)
(iii) \(\tan \theta=\frac{\text { Side opposite to } \angle A}{\text { Side adjacent to } \angle A}=\frac{\text { Perpendicular }}{\text { Base }}=\frac{B C}{A B}\)
(iv) \({\text {cosec}}\, \theta=\frac{1}{\sin \theta}=\frac{\text { Hypotenuse }}{\text { Side opposite to } \angle A}=\frac{\text { Hypotenuse }}{ \text { Perpendicular }}=\frac{A C}{B C}\)
(v) \(\sec \theta=\frac{1}{\cos \theta}=\frac{\text { Hypotenuse }}{\text { Side adjacent to } \angle A}=\frac{\text { Hypotenuse }}{\text { Base }}=\frac{A C}{A B}\)
(vi) \(\cot \theta=\frac{1}{\tan \theta}=\frac{\text { Side adjacent to } \angle A}{\text { Side opposite to } \angle A}=\frac{\text { Base }}{\text { Perpendicular }}=\frac{A B}{A C}\)

Relationship Between Trigonometric Ratios

The following is a list of some of the trigonometric ratios and their relationships.

1. \(\tan A=\frac{\sin A}{\cos A}\)
2. \(\cot A=\frac{\cos A}{\sin A} \Rightarrow \cot A=\frac{1}{\tan A}\)
3. \(\operatorname{cosec} A=\frac{1}{\sin A} \Rightarrow \sin A=\frac{1}{\operatorname{cosec} A}\)
4. \(\sec A=\frac{1}{\cos A} \Rightarrow \cos A=\frac{1}{\sec A}\)

We notice that \(\sin \theta\) is a reciprocal of \(\theta, \cos \theta\) is a reciprocal of \(\sec \theta\),
\(\tan \theta\) is a reciprocal of \(\cot \theta\) and vice-versa.

Trigonometric Identities

An equation is called an identity when it is true for all values of the variables involved. An equation involving trigonometric ratios of an angle is called a trigonometric identity if it is valid for all angle(s) values involved.

The identities between the trigonometric ratios are listed below.

1. \(\sin ^{2} \theta+\cos ^{2} \theta=1\)
2. \(\sec ^{2} \theta-\tan ^{2} \theta=1\)
3. \(\operatorname{cosec}^{2} \theta-\cot ^{2} \theta=1\)

Trigonometric Ratios of Some Specific Angles

In this section, we will learn the values of the trigonometric ratios of the angles \(0^{\circ}, 30^{\circ}, 45^{\circ}, 60^{\circ}\) and \(90^{\circ}\) which are derived using Pythagoras theorem in a right-angled triangle.

Below is the table of all the values of trigonometric ratios of \(0^{\circ}, 30^{\circ}, 45^{\circ}, 60^{\circ}\), \(90^{\circ}, 180^{\circ}, 270^{\circ}\), and \(360^{\circ}\)

Steps to Create the Trigonometric Table

Step 1: Create a table with the angles \(0^{\circ}, 30^{\circ}, 45^{\circ}, 60^{\circ}\), and \(90^{\circ}\) on the top row and all trigonometric functions \(\sin , \cos , \tan , \operatorname{cosec}, \mathrm{sec}\), and cot in the first column.

Step 2: Determine the value of \(\sin\).

Write the angles \(0^{\circ}, 30^{\circ}, 45^{\circ}, 60^{\circ}, 90^{\circ}\) in ascending order and assign them values \(0,1,2,3,4\) according to the order.

So, \(0^{\circ} \rightarrow 0 ; 30^{\circ} \rightarrow 1 ; 45^{\circ} \rightarrow 2 ; 60^{\circ} \rightarrow 3 ; 90^{\circ} \rightarrow 4\)

Then divide the values by \(4\) and square root the entire value.

\(0^{\circ} \rightarrow \sqrt{\frac{0}{4}}=0 ; 30^{\circ} \rightarrow \sqrt{\frac{1}{4}}=\frac{1}{2} ; 45^{\circ} \rightarrow \sqrt{\frac{2}{4}}=\frac{1}{\sqrt{2}} ; 60^{\circ} \rightarrow \sqrt{\frac{3}{4}}=\frac{\sqrt{3}}{2} ; 90^{\circ} \rightarrow \sqrt{\frac{4}{4}}=1\)

This gives the values of sine for these five angles.
Now for the remaining three use:
\(\sin \left(180^{\circ}-x\right)=\sin x\)
\(\sin \left(180^{\circ}+x\right)=-\sin x\)
\(\sin \left(360^{\circ}-x\right)=-\sin x\)
This means,
\(\sin \left(180^{\circ}-0^{\circ}\right)=\sin 0^{\circ}\)
\(\sin \left(180^{\circ}+90^{\circ}\right)=-\sin 90^{\circ}\)
\(\sin \left(360^{\circ}-0^{\circ}\right)=-\sin 0^{\circ}\)
So, the values of \(\sin x\) is as given below.

Step 3: Determine the value of \(\cos\).
\(\sin \left(90^{\circ}-x\right)=\cos x\)
To find values for \(\cos x\), use this formula. For example, equals \(\sin \left(90^{\circ}-45^{\circ}\right)=\cos 45^{\circ}\),
\(\sin \left(90^{\circ}-30^{\circ}\right)=\cos 30^{\circ}\) and vice versa.
You can quickly determine the value of the \(\cos\) function by using this method:

Step 4: Determine the value of \(\tan\).
We know that
\(\frac{{\sin }}{{\cos }} = \tan \)
Divide the value of \(\sin\) at \(0^{\circ}\) by the value of \(\cos\) at \(0^{\circ}\) to get the value of \(\tan\) at \(0^{\circ}\). Take a look at the sample below.
\(\tan 0^{\circ}=\frac{0}{1}=0\)
In the same way, the table would be.

Step 5: Determine the value of \(\cot\).
The reciprocal of \(\tan\) is equal to the value of \(\cot\). Divide \(1\) by the value of \(\tan\) at \(0^{\circ}\) to get the value of \(\cot\) at \(0^{\circ}\). So, \(\cot 0^{\circ}=\frac{1}{0}=\infty\) or not defined.
In the same way, a \(\cot\) table is provided below.

Step 6: Determine the value of \({\text{cosec}}\).
The reciprocal of \(\sin\) at \(0^{\circ}\) is the value of \({\text{cosec}}\) at \(0^{\circ}\).
\(\operatorname{cosec} 0^{\circ}=\frac{1}{0}=\infty\) or Not defined
In the same way, a table for \({\text{cosec}}\) is provided below.

Step 7: Determine the value of \(\sec\).
All reciprocal values of \(\cos\) can be used to calculate the value of \(\sec\). The value of \(\sec\) on \(0^{\circ}\) is the inverse of the value of \(\cos\) on \(0^{\circ}\). As a result, the value will be
\(\sec 0^{\circ}=\frac{1}{1}=1\).
Similarly, the table for \(\sec\) is shown below.

Trigonometric Ratios of Complementary Angles

Two angles are said to be complementary if their sum is \(90^{\circ}\).

In the \(\triangle A B C, \angle A+\angle C=90^{\circ}\).
\(\sin \theta=\frac{B C}{A C}, \cos \theta=\frac{A B}{A C}, \tan \theta=\frac{B C}{A B}, \theta=\frac{A C}{B C}, \sec \theta=\frac{A C}{A B}, \cot \theta=\frac{A B}{B C}\ldots…(i)\)

Now, let us write trigonometric ratios for \(\angle C=90^{\circ}-\theta\).

The opposite side of \(90^{\circ}-\theta\) is \(A B\), and the adjacent side of \(90^{\circ}-\theta\) is \(B C\).

Therefore,

\(\sin \left(90^{\circ}-\theta\right)=\frac{A B}{A C}, \cos \left(90^{\circ}-\theta\right)=\frac{B C}{A C}, \tan \left(90^{\circ}-\theta\right)=\frac{A B}{B C}, \operatorname{cosec}\left(90^{\circ}-\theta\right)=\frac{A C}{A B}\),

\(\sec \left(90^{\circ}-\theta\right)=\frac{A C}{B C}, \cot \left(90^{\circ}-\theta\right)=\frac{B C}{A B} \ldots…(ii)\)

Comparing the ratios in \((i)\) and \((ii)\), we get

\(\sin \left(90^{\circ}-\theta\right)=\frac{A B}{A C}=\cos \theta\) and \(\cos \left(90^{\circ}-\theta\right)=\frac{B C}{A C}=\sin \theta\)

Also, \(\tan \left(90^{\circ}-\theta\right)=\frac{A B}{B C}=\cot \theta\) and \(\cot \left(90^{\circ}-\theta\right)=\frac{B C}{A B}=\tan \theta\)

\(\sec \left(90^{\circ}-\theta\right)=\frac{A C}{B C}=\theta\) and \(\operatorname{cosec}\left(90^{\circ}-\theta\right)=\frac{A C}{A B}=\sec \theta\)

So, \(\sin \left(90^{\circ}-\theta\right)=\cos \theta, \cos \left(90^{\circ}-\theta\right)=\sin \theta\)

\(\tan \left(90^{\circ}-\theta\right)=\cot \theta, \cot \left(90^{\circ}-\theta\right)=\tan \theta\)

\(\sec \left(90^{\circ}-\theta\right)=\theta, \operatorname{cosec}\left(90^{\circ}-\theta\right)=\sec \theta\)

Trigonometric Ratios of Supplementary Angles

Two angles are said to be supplementary if their sum is \(180^{\circ}\). The supplement of an angle \(\theta\) is \(180^{\circ}-\theta\).

The trigonometric ratios of supplementary angles are:

(i) \(\sin \left(180^{\circ}-\theta\right)=\sin \theta\)
(ii) \(\cos \left(180^{\circ}-\theta\right)=-\cos \theta\)
(iii) \({\text {cosec}}\,\left(180^{\circ}-\theta\right)={\text {cosec}}\,\theta\)
(iv) \(\sec \left(180^{\circ}-\theta\right)=-\sec \theta\)
(v) \(\tan \left(180^{\circ}-\theta\right)=-\tan \theta\)
(vi) \(\cot \left(180^{\circ}-\theta\right)=-\cot \theta\)

Solved Examples – Introduction to Trigonometry

Q.1. A building is at a distance of \(120 \mathrm{~m}\) from point \(A\) on the ground. Can you calculate the height of the building using the trigonometric ratios if \(\tan \tan \theta=\frac{4}{3}\) ?
Ans: The triangle formed is right-angled. Now apply the trigonometric ratio of \(\tan \tan \theta\) to calculate the height of the building.

\({\text{tan tan }}\theta\, {\text{=}}\,\frac{{{\text{Perpendicular}}}}{{{\text{Base}}}}\)
\(\frac{4}{3}=\frac{\text { Height }}{120} \mathrm{~m}\)
Height \(=\left(4 \times \frac{120}{3}\right) \text m=160 \mathrm{~m}\)
Hence, the height of the building is \(160 \mathrm{~m}\).

Q.2. Evaluate \(\cos ^{2} 13^{\circ}-\sin ^{2} 77^{\circ}\)
Ans: \(\cos ^{2} 13^{\circ}-\sin ^{2} 77^{\circ}\)
We know that, \(\cos \left(90^{\circ}-A\right)=\sin A\)
Also, \(\cos ^{2} 13^{\circ}=\left(\cos 13^{\circ}\right)^{2}\) and \(\sin ^{2} 77^{\circ}=\left(\sin 77^{\circ}\right)^{2}\)
So, \(\cos 13^{\circ}=\left[\cos \left(90^{\circ}-77^{\circ}\right)\right]^{2}=\left(\sin 77^{\circ}\right)^{2}\)
\(\Rightarrow \cos ^{2} 13^{\circ}-\sin ^{2} 77^{\circ}=\sin ^{2} 77^{\circ}-\sin ^{2} 77^{\circ}=0\)
Therefore, \(\cos ^{2} 13^{\circ}-\sin ^{2} 77^{\circ}=0\)

Q.3. Find the value of \(\frac{4}{3} \tan 45^{\circ}+2 \cos 30^{\circ}-3 \sec 30^{\circ}\)
Ans: The given expression is \(\frac{4}{3} \tan 45^{\circ}+2 \cos 30^{\circ}-3 \sec 30^{\circ}\).
Putting the value of the angles using the trigonometric table, we have
\(=\frac{4}{3}(1)+2\left(\frac{\sqrt{3}}{2}\right)^{2}-3\left(\frac{2}{\sqrt{3}}\right)^{2}\)
\(=\frac{4}{3}+2 \times \frac{3}{4}-3 \times \frac{4}{3}\)
\(=\frac{4}{3}+\frac{3}{2}-4\)
\(=\frac{-7}{6}\)

Q.4. If \(\sin (x+y)=1\) and \(\cos (x-y)=\frac{1}{2}\), find \(x\) and \(y\).
Ans: \(\sin (x+y)=1\)
\(\Rightarrow \sin (x+y)=\sin 90^{\circ},\left[\right.\)since \(\left.\sin 90^{\circ}=1\right]\)
\(\Rightarrow x+y=90^{\circ} \ldots \ldots \ldots \ldots \ldots \ldots \ldots\ (i)\)
\(\cos (x-y)=\frac{1}{2}\)
\(\Rightarrow(x-y)=\cos 60^{\circ}\)
\(\Rightarrow x-y=60^{\circ} \ldots \ldots \ldots \ldots \ldots \ldots \ldots\ (ii)\)
Adding, \((i)\) and \((ii)\), we get
\(x+y=90^{\circ}\)
\(x-y=60^{\circ}\)
\(2 x=150^{\circ}\)
\(x=75^{\circ}\), [Dividing both sides by \(2\)]
Putting the value of \(x=75^{\circ}\) in \((i)\) we get,
\(75^{\circ}+y=90^{\circ}\)
Subtract \(75^{\circ}\) from both sides
\(75^{\circ}+y=90\)
\(y=15^{\circ}\)
Therefore, \(x=75^{\circ}\) and \(y=15^{\circ}\).

Q.5. If \(\cot A=\tan B\), prove that \(A+B=90^{\circ}\).
Ans: Given: \(\cot A=\tan B\)
We know that, \(\tan B=\cot \left(90^{\circ}-B\right)\)
\(\Rightarrow \cot A=\cot \left(90^{\circ}-B\right)\)
Since, \(A\) and \(B\) are acute angles,
\(A=90^{\circ}-B\)
\(\Rightarrow A+B=90^{\circ}\)
Hence, proved.

Summary

In the above article, we have learned the meaning of trigonometry and trigonometric ratios of some specific angles. We have also known the relationship between trigonometric ratios and the identities between these ratios.

Also, we have learned to find the trigonometric ratios of complementary angles, supplementary angles and solved some example problems based on them.

Frequently Asked Questions (FAQs) – Introduction to Trigonometry

The most frequently asked questions about Introduction to Trigonometry are answered here:

We hope you find this detailed article on the introduction of trigonometry useful. If you have any doubts or queries on this topic, feel to ask us in the comment section, and we will be more than happy to assist you.