Trigonometry Formulas

Trigonometry Formulas – Trigonometry is a branch of Mathematics that deals explicitly with the angles of a triangle and tries to see the relationship between each element of the triangle which is three sides and three angles. Trigonometry also deals with the relationship of each angle of triangles with circles and is very specifically used in the branch of science and engineering. 

This list of trigonometry formulae is organised by trigonometric ratios such as sine, cosine, tangent, cotangent, secant, and cosecant. These formulae are used to solve a variety of trigonometric issues.

Trigonometry is one of the most significant concepts in mathematics. Trigonometry is essentially the study of triangles, with ‘trigon’ meaning triangle and’metry’ meaning measurement.

Trigonometry Formulas

Trigonometry formulae are mathematical equations that describe the angles and sides of a right triangle. They are used in trigonometry to answer a variety of issues involving angles, distances, and height. Using these calculations, one may determine the missing side or angle of a right triangle.

In addition to fundamental formulae like the Pythagorean theorem, there are several trigonometric identities and formulas that may be used to simplify expressions, solve equations, and calculate integrals. These formulae are vital tools for engineers, mathematicians, and scientists working in a wide range of fields.

Trigonometry Formulas List

The formulas or basically the relations that we are going to learn and understand are the followings

• • Basic Formulas
  • Reciprocal Identities
  • Trigonometry Table
  • Periodic Identities
  • Co-function Identities
  • Sum and Difference Identities
  • Double Angle Identities
  • Triple Angle Identities
  • Half Angle Identities
  • Product Identities
  • Sum to Product Identities
  • Inverse Trigonometry Formulas

Basic Trigonometric Function Formulas

In trigonometry, there are 6 basic relations defined with 3 sides and 3 angles. Let’s have a look at them 

• sin θ = Opposite Side/Hypotenuse
• cos θ = Adjacent Side/Hypotenuse
• tan θ = Opposite Side/Adjacent Side
• sec θ = Hypotenuse/Adjacent Side
• cosec θ = Hypotenuse/Opposite Side
• cot θ = Adjacent Side/Opposite Side

Here the opposite is the side opposite to the angle taken into consideration and adjacent is the side adjacent to the same angle considered.

Reciprocal Identities

All these relations are when taken reciprocals of produce a different set of results. Before moving further, try taking reciprocals of each and find out what they become. 

For example

• sin θ = Opposite Side/Hypotenuse

1/sin θ= hypotenuse/ opposite

Which is following this same ratio? Cosecant, so let’s see the reciprocal identities for all the angles. 

• cosec θ = 1/sin θ
• sec θ = 1/cos θ
• cot θ = 1/tan θ
• sin θ = 1/cosec θ
• cos θ = 1/sec θ
• tan θ = 1/cot θ

Trigonometry Table

These are the common values of angles that are used for solving problems. Memorising these just saves some time, so here is a table. 

Periodicity Identities (in Radians)

Let’s look at the above table of values and let’s see the values of Sin in 0 and 360. As we see they are the same. So basically, we see these values repeat after a particular set of intervals. Hence we call these functions periodic. 

Remember these are in radians. 

• sin (π/2 – A) = cos A & cos (π/2 – A) = sin A
• sin (π/2 + A) = cos A & cos (π/2 + A) = – sin A
• sin (3π/2 – A) = – cos A & cos (3π/2 – A) = – sin A
• sin (3π/2 + A) = – cos A & cos (3π/2 + A) = sin A
• sin (π – A) = sin A & cos (π – A) = – cos A
• sin (π + A) = – sin A & cos (π + A) = – cos A
• sin (2π – A) = – sin A & cos (2π – A) = cos A
• sin (2π + A) = sin A & cos (2π + A) = cos A

Cofunction Identities (in Degrees)

Co function is basically the relationship of one ratio with another one. Remember they’re in degrees. 

• sin(90°−x) = cos x
• cos(90°−x) = sin x
• tan(90°−x) = cot x
• cot(90°−x) = tan x
• sec(90°−x) = cosec x
• cosec(90°−x) = sec x

Sum & Difference Identities

• sin(x+y) = sin(x)cos(y)+cos(x)sin(y)
• cos(x+y) = cos(x)cos(y)–sin(x)sin(y)
• tan(x+y) = (tan x + tan y)/ (1−tan x •tan y)
• sin(x–y) = sin(x)cos(y)–cos(x)sin(y)
• cos(x–y) = cos(x)cos(y) + sin(x)sin(y)
• tan(x−y) = (tan x–tan y)/ (1+tan x • tan y)

These are used when we have an angle on which we have to use the function of Sin, which can be defined by two standard angles or if we can define any unknown angle by the sum or difference of any standard angle. 

Double Angle Identities
These are used when the given angle is the double of any standard angle.

• sin(2x) = 2sin(x) • cos(x) = [2tan x/(1+tan2 x)]
• cos(2x) = cos2(x)–sin2(x) = [(1-tan2 x)/(1+tan2 x)]
• cos(2x) = 2cos2(x)−1 = 1–2sin2(x)
• tan(2x) = [2tan(x)]/ [1−tan2(x)]
• sec (2x) = sec2 x/(2-sec2 x)
• csc (2x) = (sec x. csc x)/2

Triple Angle Identities

• Sin 3x = 3sin x – 4sin3x
• Cos 3x = 4cos3x-3cos x
• Tan 3x = [3tanx-tan3x]/[1-3tan2x]

For other ratios, we just find the answer to these and invert them.

Half Angle Identities

These are used when the given angle is half of the standard angle. Also for other ratios, we invert the answer found by these ratios.in these, as we see, we can also find the ratio of tan with sin and cos.

 Product identities

These are used when we need to find the product of functions of different angles. 

 Sum to Product Identities

1. Sinα±sinβ=2sin12(α±β)Cos12(α∓β)
2.  Cosα+Cosβ=2Cos12(α+β)Cos12(α−β)
3. Cosα–Cosβ=−2sin(α+β)/2sin(α–β)/2

Inverse Trigonometry Formulas with Their Domain and Range

Remember, inverse trigonometry is different from reciprocals. Reciprocal is when we put the entire function from numerator to denominator and vice versa, which is sin x and 1/sin x. On the other hand, the inverse is basically answering the question, let’s take the example of sin inverse, what angle of Sin is the given value of opposite/hypotenuse.

So the formulas are 

• sin-1 (–x) = – sin-1 x
• cos-1 (–x) = π – cos-1 x
• tan-1 (–x) = – tan-1 x
• cosec-1 (–x) = – cosec-1 x
• sec-1 (–x) = π – sec-1 x
• cot-1 (–x) = π – cot-1 

θ = sin−1

(x) is equivalent to x = sin θ

θ = cos−1

(x) is equivalent to x = cos θ

θ = tan−1

(x) is equivalent to x = tan θ

 These properties hold for x in the domain and θ in

the range

sin(sin−1(x)) = x

cos(cos−1(x)) = x

tan(tan−1(x)) = x

sin−1(sin(θ)) = θ

cos−1(cos(θ)) = θ

tan−1(tan(θ)) = θ

Use the same logic as above to see why the domains and the ranges are there as these.

What is the Sin 3x Formula?

Sine 3x is basically the sine of triple the angle that can be permissible in any RAT

Remember Sin 3x will also be between -1 and 1. The range is true regardless of any angle put into this function. 

The formula for Sin 3x is 

Sin 3x = 3sin x – 4sin3x

Trigonometry Solved Problems

Q.1: Find the value of Calculate sin75° sin15° 

Solution: As given,

sin75° sin15° 

= sin(90° −15° )sin15° 

= cos15° sin15° 

= sin30° /2 [applying sin2x=2sin(x)cos(x)]

= 1/4

Q.2: What is the value of (sin30° + cos30°) – (sin 60° + cos60°)?

Solution: 

Given,

(sin30° + cos30°) – (sin 60° + cos60°)

= ½ + √3/2 – √3/2 – ½

= 0

Q.3: If cos A = 4/5, then tan A = ?

Solution: 

Given, 

Cos A = ⅘

As we know, from trigonometry identities,

1+tan2A = sec2A

sec2A – 1 = tan2A

(1/cos2A) -1 = tan2A

Putting the value of cos A = ⅘.

(5/4)2 – 1 = tan2 A

tan2A = 9/16

tan A = ¾

Q.4: A person 100 metres from the base of a tree, observes that the angle between the ground and the top of the tree is 18 degrees. Estimate the height h of the tree to the nearest tenth of a metre.

Solution:

• Use the tangent
  tan(18o) = h / 100
• Solve for h to obtain
  h = 100 tan(18o) = 32.5 metres.

Q.5: The angle of elevation of a hot air balloon, climbing vertically, changes from 25 degrees at 10:00 am to 60 degrees at 10:02 am. The point of observation of the angle of elevation is situated 300 metres away from the take-off point. What is the upward speed, assumed constant, of the balloon? Give the answer in metres per second and round to two decimal places.

Solution:

• Use the tangent to write
  tan(25o) = h1 / 300
  and
  tan(60o) = (h1 + h2) / 300
• Solve for h1 and h2
  h1 = 300 tan(tan(25o))
  and
  h1 + h2 = 300 tan(60o)
• Use the last two equations to find h2
  h2 = 300 [ tan(60o) – tan(25o) ]
• If it takes the balloon 2 minutes (10:00 to 10:02) to climb h2, the the upward speed S is given by
  S = h2 / 2 minutes
  = 300 [ tan(60o) – tan(25o) ] / (2 * 60) = 3.16 m/sec