What will you learn in this article
- Trigonometric Identities
- Heights and Distance problems related to Trigonometry
- Measurement of angles and constant number π
- Relation between Arc and Angle
Before continuing with this topic, make sure you have read Higher Maths | Trigonometry #1 | Basic Concepts to understand the basic understanding of trigonometry and trigonometric ratios of some specific angles and complementary angles
Trigonometric Identities are basically equations involving trigonometric ratios of an angle. Trigonometric Identities shows a relationship between all the trigonometric ratios of an angle.
There are three trigonometric identities and these three identities are mostly used in solving trigonometric queries. They are:
- 𝚹+𝚹 = 1
- 1+ 𝚹 = 𝚹
- 𝚹+1 = 𝚹
- tan 𝚹=sin 𝚹/ cos 𝚹
- cot 𝚹= cos 𝚹/ sin 𝚹
- sin 𝚹 csc 𝚹 =cos 𝚹 sec 𝚹 =tan 𝚹 cot 𝚹 =1
- |sin 𝚹 | ≤ 1 i.e. -1 ≤ sin 𝚹 ≤1
- |cos 𝚹 | ≤1 i.e. -1 ≤ |cos 𝚹 | ≤1
- sec 𝚹 ≥1 or ≤-1
- csc 𝚹 ≥1 or ≤-1
Heights and Distance
So, to find the height and distance of one object w.r.t another object, we can find it using trigonometric ratios. We would need the following information:
- The distance at which person ‘p’ is standing at point C i.e. the distance of BC
- The angle of elevation i.e. ∠ACB
- The height of person ‘p’ if applicable or mentioned in the question. If not mentioned, then we don’t have to consider this.
Measurement of angles and constant number π
Trigonometry generally follows two systems of measuring an angle. These two systems are known as:
Let us understand these two systems in details.
1. Sexagesimal System
This system is basically defined in degrees, minutes and seconds. In simple terms, in this system, a right angled triangle is divided into 90 equal parts known as degrees. A degree is basically defined as the amount, level or extend to which something happens or is present. Now, each degree is divided into 60 equal parts known as minutes. And finally each minute is divided into 60 equal parts known as seconds. So, basically in simple terms,
- 1 right angle is equal to 90°
- 1 degree (1°) is equal to 60 minutes, which is also denoted as 60’
- 1 minute (1’) is equal to 60 seconds, which is also denoted as 60”
So, if there is a right angled triangle ABC with B being the right angle, then angle B will be given as 90°. Angle B can also be denoted as 5400 minutes and 324000 seconds.
2. Circular System
Circular system denotes the angle representation when a triangle is cut inside a given circle.
The unit of an angle in circular system is given as a radian. Radian is the angle subtended at the center of the circle by an arc whose length is equal to the radius. So, if O is the center of the circle, then
∠AOB=1 radian, where arc AB = radius
With this, there is another concept which is known as constant number π (or Pi). Pi is a constant, which is widely used in trigonometry and also in lot of other concepts that you’ll come across in your higher studies. Pi is a concept that we come across in lower classes while studying circles. We know that the area of a circle is given as πr2. In this case also, π is a constant number. Normally, π is defined as the ratio of circumference of a circle and diameter of the circle. That is,
π=circumference of the circle/diameter of the circle
The constant π is irrational and the approx. value is given as 227 or 355113. In decimal form, π is given as 3.14 (approx.).
Relation between Arc and Angles
This is a small basic concept that will help you in simplifying lot of trigonometry queries. If an arc subtends an angle in a circle at the center, then the number of radians in an angle is given as:
Where θ is the angle, ‘l’ is the arc and ‘r’ is the radius. So, l=θr
Complete the trigonometry series by reading the final article on addition and subtraction of trigonometric functions and other topics here.