Trigonometry is one of the most intriguing branches of mathematics. It deals with the relationships between the angles and sides of a right-angled triangle, and the applications of these in calculating heights and distances. Here, we delve into the history, trigonometric functions, calculations, and applications of trigonometry, in addition to solving some problems associated with it.
Table of Contents:
Important Trigonometry Angles
List of Trigonometry Formulas
Examples in Real Life
Problems and Solutions
Introduction to Trigonometry
The study of triangles dates back to the 2nd millennium, found in the Babylonian and Egyptian mathematical records. However, the roots of modern trigonometry can be traced back to the Greeks, with Hipparchus credited with creating the first table for trigonometric functions. Later on, Indian mathematicians like Aryabhata I contributed further by giving the first table of sines in trigonometry. As we moved into the Middle Ages, Arab mathematicians like Al-Khwarizmi added further to our understanding of this field. This knowledge transferred to the Latin West during Renaissance and from there it moved on to the West, where great mathematicians like Isaac Newton and Leonhard Euler laid the foundation for modern trigonometry.The word ‘Trigonometry’ comes from the Greek terms ‘trigōnon’ for ‘triangle’ and ‘metron’ for ‘measure’. In simple words, it is the study of relationships that deal with angles, lengths and heights of triangles and relations between different parts of circles and other geometrical figures. Applications of this branch of mathematics are found in engineering, astronomy, physics and architectural design.
A right-angle triangle is a special triangle in which one angle is 90° and the other two are less than 90°. Furthermore, each side of the right angle triangle has a name:
Hypotenuse: It is the largest side of the triangle. Also, it is opposite the right angle of the triangle.
Base: The side on which the right angle triangle stands is known as its base. Moreover, any of the two sides other than the hypotenuse can be chosen as the base for performing the calculation.
Perpendicular: It is the side perpendicular to the base of the right-angled triangle.
Trigonometry formulas are based on the relationships between these three sides and trigonometry angles.
Based on the relationships between the lengths of these sides of the triangle and the angles of the triangle, there are six trigonometric functions – sine, cosine, secant, cosecant, tangent, and cotangent. All formulae are associated with these six functions. Find the list of trigonometry formulas here. These are trigonometry basics you need to know for solving related numericals.
The six trigonometric ratios of a right angle triangle are Sin, Cos, Tan, Cosec, Sec and Cot. They stand for Sine, Cosine, Tangent, Cosecant, Secant, and Cotangent respectively. The relationship between the measurement of the trigonometry angles and the length of the side of the right triangle are called the trigonometric ratios. Trigonometry ratios are applicable only for a right-angle triangle and are used for deriving trigonometry formulas.
As mentioned earlier, a right-angled triangle has a hypotenuse, a base, and a perpendicular. The longest side is known as the hypotenuse, the other side which is opposite to the angle is called perpendicular and the third side is called the base.
Trigonometric ratios are the ratios of sides of a right-angle triangle. Consider a right-angle triangle ABC, right-angled at C. In that case, side AB will be the hypotenuse. Also, if we chose AC as the base and BC as the perpendicular. Then, for ∠BAC, value of sinθ = Perpendicular/ hypotenuse = BC/AB.
For ease of understanding, we can name the sides of the triangle as follows based on the angle we are considering:
Adjacent: It is the side adjacent to the angle being considered.
Opposite: It is the side opposite to the angle being considered.
Hypotenuse: It is the side opposite to the right angle of the triangle (or the largest side).
Accordingly, the trigonometry ratios are written as follows based on the corresponding angles:
- sine or sinθ = Perpendicular/Hypotenuse = Opposite/Hypotenuse
- cosine or cosθ = Base/ Hypotenuse = Adjacent/Hypotenuse
- tangent or tanθ = Perpendicular/Base = Opposite/Adjacent.
We can also take the inverse of these trigonometric functions as follows:
- Reciprocal of sine function: cosecant or cosecθ = Hypotenuse/Perpendicular = Hypotenuse/Opposite
- Reciprocal of cosine function: secant or secθ = Hypotenuse/Base = Hypotenuse/ Adjacent
- Reciprocal of tangent function: cotangent or cotθ = Base/Perpendicular = Adjacent/Opposite
Trigonometry basics cover these six important functions as they are applied to solve problems based on heights and distances.
The six trigonometric functions
|S. No.||Property||Mathematical Value|
|1||sin A||P/H (Perpendicular / Hypotenuse)|
|2||cos A||B/H (Base / Hypotenuse)|
|3||tan A||P/B (Perpendicular / Base)|
|4||cot A||B/P (Base / Perpendicular)|
|5||cosec A||H/P (Hypotenuse / Perpendicular)|
|6||sec A||H/B (Hypotenuse / Base)|
Relation between trigonometric identities
|1||tan A||sin A/cos A|
|2||cot A||cos A/sin A|
|3||cosec A||1/sin A|
|4||sec A||1/cos A|
The six trigonometric functions can be listed in a table, called the trigonometric table. This table gives the numerical values of the different trigonometry ratios and helps solve problems using formulas.
|tan A||0||13||1||3||Not defined|
|cot A||Not defined||3||1||13||0|
|sec A||1||23||2||2||Not defined|
|cosec A||Not defined||2||2||23||1|
The trigonometric table has the trigonometry ratios for all the functions – sine, cosine, tangent, cosecant, secant, cotangent. All these ratios are interrelated as you can see in the table above. The numerical values of these functions are derived and considered as the standard angle values. The values in this table are used to solve problems using formulas. You will be able to find a list of trigonometry formulas in an upcoming section.
Important Trigonometry Angles
The most commonly used trigonometry angles are 0°, 30°, 45°, 60° and 90°. Apart from these, 180°, 270°, and 360°angles are also applied. A trigonometric angle could be any one of the two acute angles in a right-angled triangle. Each of these angles is defined by the trigonometric ratios associated with it. For example,
- θ = sin-1 (Perpendicular/Hypotenuse); (P/H)
- θ = cos-1 (Base/Hypotenuse); (B/H)
- θ = tan-1 (Perpendicular/Base); (P/B)
We can define trigonometry angles as the angles corresponding to ratios of trigonometric functions. These angles form an important part of all trigonometry formulas.
List of Trigonometry Formulas
Trigonometric identities or formulas are formulas involving trigonometric functions. These trigonometric identities are true for all values of the variables. Given below is a list of trigonometry formulas:
The basic trigonometry formulas based on the Pythagoras Theorem are listed here. You can use the basic definition and Pythagoras theorem to prove these.
sin2θ + cos2θ = 1
1 + tan2θ = sec2θ
1 + cot2θ = cosec2θ
These three formulae or identities are important trigonometry basics as other formulae can be derived from them.
As the name suggests, these relations involve two trigonometric ratios which are connected by inverse relations between them. For example,
sin θ = 1/ cosec θ or sin θ x cosec θ = 1
cos θ = 1/ sec θ or cos θ x sec θ = 1
tan θ = 1/cot θ or tan θ x cot θ = 1
Again, as the name suggests, quotient relations involve three trigonometric ratios; where one is the quotient obtained after division operation between the other two. For example,
tan θ = sin θ /cos θ
cot θ = cos θ / sin θ
Trigonometry ratios for negative trigonometry angles can be derived using the circular concept of negative angles and can be derived using cartesian notation and conventions.
Periodicity and periodic identities
The basic concept of trigonometric functions is based on the repetition of the values of sine, cos and tan after 360⁰ due to their periodic nature. If n is an integer and in radians (if in degrees the replace with 360).
sin (θ + 2n) = sin θ
cos (θ + 2n) = cos θ
tan (θ + 2n) = tan θ
csc (θ + 2n) = csc θ
sec (θ + 2n) = sec θ
cot (θ + 2n) = cot θ
Reduction trigonometry formulas
If the angles are given in any of the four quadrants then the angle can be reduced to the equivalent first quadrant by changing signs and trigonometric ratios. Here’s the list of trigonometry formulas for each quadrant.
sin (2- θ) = cos θ
cos (2- θ) = sin θ
tan (2- θ) = cot θ
csc (2- θ) = sec θ
sec (2- θ) = csc θ
cot (2- θ) = tan θ
sin (- θ) = sin θ
cos (- θ) = – cos θ
tan (- θ) = – tan θ
csc (- θ) = csc θ
sec (- θ) = – sec θ
cot (- θ) = – cot θ
sin (+ θ) = – sin θ
cos (+ θ) = – cos θ
tan (+ θ) = tan θ
csc (+ θ) = – csc θ
sec (+ θ) = – sec θ
cot (+ θ) = cot θ
sin (2- θ) = – sin θ
cos (2- θ) = cos θ
tan (2- θ) = – tan θ
csc (2- θ) = – csc θ
sec (2- θ) = sec θ
cot (2- θ) = – cot θ
Sum to product rules
Some trigonometry formulas related to the sum and difference of two angles can be listed as follows:
sin (A+B) = sin A cos B + cos A sin B
sin (A-B) = sin A cos B – cos A sin B
cos (A+B) = cos A cos B – sin A sin B
cos (A-B) = cos A cos B + sin A sin B
tan (A+B) = (tan A + tan B)/(1 – tan A tan B)
tan (A-B) = (tan A – tan B)/(1 + tan A tan B)
a/sinA = b/sinB = c/sinC
sinA/a = sinB/b = sinC/c
a/b = sinA/sinB; a/c = sinA/sinC; b/c = sinB/sinC
a2 = b2 + c2 – 2bc·cosA
b2 = c2 + a2 – 2ca·cosB
c2 = a2 + b2 – 2ab·cosC
They are sometimes denoted with a-1 in the superscript of the trigonometry ratios and sometimes also denoted using arc as a prefix, for example, sin-1, cos-1, arctan etc.
|Inverse Trigonometric Functions||Formulas|
|Arcsine||sin-1(-x) = -sin-1(x), x ∈ [-1, 1]|
|Arccosine||cos-1(-x) = π -cos-1(x), x ∈ [-1, 1]|
|Arctangent||tan-1(-x) = -tan-1(x), x ∈ R|
|Arccotangent||cot-1(-x) = π – cot-1(x), x ∈ R|
|Arcsecant||sec-1(-x) = π -sec-1(x), |x| ≥ 1|
|Arccosecant||cosec-1(-x) = -cosec-1(x), |x| ≥ 1|
Complimentary trigonometry angles
sin (90°- θ) = cos θ
cos (90°- θ) = sin θ
cosec (90°- θ) = sec θ
sec (90°- θ) = cosec θ
tan (90°- θ) = cot θ
cot (90°- θ) = tan θ
Supplementary trigonometry angles
sin (180°- θ) = sinθ
cos (180°- θ) = -cos θ
cosec (180°- θ) = cosec θ
sec (180°- θ)= -sec θ
tan (180°- θ) = -tan θ
cot (180°- θ) = -cot θ
Examples of Trigonometry in Real Life
Trigonometry can be applied to numerous aspects in real life. Here’s an example of how trigonometric functions can be applied to real life. Let’s consider that the sun’s rays are casting the shadow of a tree and a stick at the same time. The tree, its shadow, and the sun’s rays form a right-angled triangle as shown.
Similarly, a right-angled triangle is formed in the case of the stick and its shadow. Now, we can compare the trigonometry ratios of original lengths and length of the shadows as shown.
With the help of trigonometry ratios and trigonometric formulae, we can calculate the length of the stick, the length of the tree and much more using the list of trigonometry formulas in the previous section.
Here’s another example of the application of trigonometric ratios:
Consider a tank and an enemy plane in its vicinity. The soldier in the tank can guess his distance on the ground by a device present with him and also the plane’s height from the ground.
Using this information, he can estimate the angle between him and the plane with respect to the ground, so that he can shoot the plane.
This calculation is also possible using trigonometric functions.
Applications of Trigonometry
Trigonometry has many uses such as to roof a house, to make roof inclined (in case of single individual bungalows) and the height of the roof in a building, etc. In addition, the aviation and naval industry use it. Moreover, cartographers use it in creating maps and it has applications in satellite systems. Trigonometric ratios can be used to find heights and distances between near and far objects. It is used in oceanography, flight engineering, and even archeology.
Trigonometry Problems and Solutions
You can apply the trigonometry basics learnt in previous sections to each of these questions.
Question 1: Consider a right angle triangle ABC right angled at C. If the hypotenuse = AB = 5cm, perpendicular =BC = 4cm and base = AC = 3cm. Then, for ∠BAC=θ, calculate the value of sinθ, cosθ and tanθ.
Answer: For right angle triangle ABC,
sinθ = Perpendicular/ Hypotenuse =4/5
cosθ = Base/ Hypotenuse =3/5
tanθ = Perpendicular/Base = 4/3
Question 2: In a right triangle with sides a and b, hypotenuse H, the altitude p drawn on hypotenuse is, then ab = ?
a2 + b2 = H (equation 1)
sin (90-θ) =PQ
cos θ = PQ
sin θ = pb
As we know that sin2 x + cos2 x = 1
P2a2+ P2b2= 1
Solving and then substituting equation 1 gives: PH = ab
Question 3: The value of sin 15° is?
= sin (45° – 30°)
= sin 45° cos 30° – cos 45° sin 30°
= 12. 32- 12. 12
Question 4: A vertical tower stands on a horizontal plane and is surmounted by a vertical flag staff of height 5 meters. At point on the plane, the angle of elevation of the bottom and top of the flag staff are respectively 30∘ and 60∘. What is the height of the tower?
Answer: Let AB be the tower of height h meter and BC be the height of flag staff surmounted on the tower. Let the point of the plane be D at a distance m meter from the foot of the tower.
tan 30° = ABBD
x = 3h (equation 1)
tan 60° = ACAD
x= 5+h3 (equation 1)
3h = 5 +h
Solving for h gives 2.5m
Question 1: What are trigonometric identities?
Answer: Trigonometric identities are used to calculate the value of the sides and angles of right angled triangles. Furthermore, these identities are also beneficial in practical life circumstances, for instance, calculating the height of a building and so on. They form the trigonometry basics, as they are used in calculations and problem solving.
Question 2: What are quotient relations?
Answer: As evident from the name, quotient relations comprise three trigonometry ratios. One is the quotient we get after division operation between the other two. For instance, tan θ = sin θ /cos θ and cot θ = cos θ / sin θ.
Question 3: Who is the father of trigonometry?
Answer: Hipparchus is referred to as the father of trigonometry and was a Greek astronomer. It is a branch of mathematics that studies the angles of sides of triangles. Moreover, Hipparchus also created the first accurate star map.
Question 4: What are the laws of trigonometry?
Answer: The trigonometric laws of identity comprise these significant reciprocal equalities. The tangent is the sine we divide by the cosine. The cotangent is equal to one over the tangent, or the cosine we divide by the sine. The secant is equal to one over the cosine and the cosecant is equal to one over the sine.
Question 5: What are the 3 trigonometry ratios?
Answer: The 3 trigonometry ratios are sine, cosine, and tangent. We can find out the sine (or cosine or tangent) of either of the known trigonometry angles.
Question 6: What is the use of trigonometry ratios in a right angle triangle?
Answer: Trigonometry ratios apply to a right angle triangle only. It is a special triangle in which one angle is 90° and the other two are less than 90°. Also, each side of the triangle has a name. They are hypotenuse, perpendicular, and base.
Question 7: What is the tangent ratio?
Answer: It is a tool we use with right triangles. It lets us find the lengths of the sides when the degrees of their angles are given. Also, we can use it to find out the degrees of its angle when the lengths of two of the sides are given.
Question 8: Is trigonometry applicable to all the triangles?
Answer: Basically, trigonometry functions are defined for right triangles, but they also work for non-right triangles. Furthermore, we know them as the law of sines and law of cosines. However, they work as follows for any triangle:
These are: Sin (a)/A = Sin (b)/ B = Sin (c)/ C. Here c equals 90 degrees, and then we can reduce it to Pythagorean’s theorem.
Here, we have covered trigonometry basics, list of trigonometry formulas, functions, angles and ratios. It is a very important subject, especially if you intend to study subjects related to engineering and architecture.