Learn Trigonometry: A beginner’s guide to Trigonometry.

The word Trigonometry is derived from the Greek words trigonon meaning “triangle” and metron meaning “measure”. Trigonometry is a branch of mathematics that studies the calculations of triangles and its relationship of lengths, heights, and angles. The application of trigonometry is vast and applied in various fields like physics, architecture, astronomy, engineering, crime scene investigation, oceanography, game development, developing computer music, medical engineering, phonetics, and many more. 

We make use of trigonometry in our day to day life too! For example, The GPS that you use for navigation while driving uses trigonometry for north south east west directions and guides you to your destination with the help of a compass. It pinpoints your location and helps in navigation. 

So, here’s a complete beginner’s guide for students of Class 10th, 11th, and 12th to give a clear understanding of the concerned topics and syllabus. The presented step-by-step solutions for solving problems will make it easier for students to refer from in case of any doubt. The students can also use it for quick revision and practice for competitive exams like JEE, Olympiads, government recruitment exams, and more. 

Trigonometric Ratios: 

For us to have a clear understanding of trigonometry, we need to first learn in detail about its functions (ratios). The trigonometric ratios are sine, cosine, tangent, and their reciprocals- cosecant, secant, and cotangent. The six ratios are abbreviated as- sin, cos, tan, cosec, sec, and cot.  

The functions are referred to as ratios because they are expressed in relation to the sides of a right-angled triangle for a specific angle θ. 

Note: Although trigonometric functions are specifically defined for the right-angled triangles, there still laws of cosines and sines that are applied for non-right-angled triangles. 

To know the values of the trigonometric ratios, you must refer to the trigonometry table. The table consists of the values of different trigonometric ratios for standard angles- 0°, 30°, 45°, 60°, and 90°.

Angles in Degrees	0°	30°	45°	60°	90°
Angles in Radians	0°	π/6	π/4	π/3	π/2
sin	0	1/2	1/√2	√3/2	0
cos	1	√3/2	1/√2	1/2	0
tan	0	1/√3	1	√3	N/A
cot	N/A	√3	1	1/√3	0
sec	1	2/√3	√2	2	N/A
cosec	N/A	2	√2	2/√3	1

The trigonometric ratios values for standard angles- 180°, 270°, and 360°.

Angles in Degrees	180°	270°	360°
Angles in Radians	π	3π/2	2π
sin	0	-1	0
cos	-1	0	1
tan	0	Undefined	1
cot	Undefined	0	Undefined
sec	-1	Undefined	1
cosec	Undefined	-1	Undefined

Understanding Trigonometric ratios:

Let us look at this diagram of the right-angled triangle,

 

The triangle has three sides:
1)Base: horizontal to the plane
2)Hypotenuse: longest side and opposite to right-angled vertex
3) Perpendicular: the side that makes the 90-degree angle with the base.
(Angle X or θ: angle made by Base and Hypotenuse)

Formulas that are generated with this:

• Sine of angle X = sinX = PerpendicularHypotenuse
• Cosine of angle X = cos X = BaseHypotenuse
• Tangent of angle X = tan X = PerpendicularBase

Reciprocal relationship between different trigonometric ratios:

• Cotangent of angle X = cot X = 1tanX = BasePerpendicular
• Cosecant of angle X = cosec X = 1sinX = HypotenusePerpendicular
• Secant of angle X = sec X = 1cosX = HypotenuseBase

Note:
1) If you consider the third angle between Hypotenuse and Perpendicular, then the Base becomes Perpendicular and vice versa. The values should be added accordingly for calculating the trigonometric ratios of that angle.

2) In order to remember what the trigonometric ratios of which sides correspond to which trigonometric function, use a mnemonic.
For example,
S of Some = Sine
P of People = Perpendicular
H of Have = Hypotenuse
(This gives sinX (S) = PerpendicularHypotenuse)

C of Curly ↠ Cosine
B of Black ↠ Base
H of Hair ↠ Hypotenuse
(This gives cosX (C) = BaseHypotenuse)

T of Thickly ↠ Tangent
P of Plastered ↠ Perpendicular
B of Back ↠ Base
(This gives tanX (T) = PerpendicularBase)

Cosecant, secant, and contingent are the reciprocals of sine, cosine, and tangent.

Example :- In triangle PQR, right angled at Q, where PQ is 50 cm, QR is 18 cm. Find PR .

Solution :- PQ = 50cm ; QR = 18 cm and let PR is x

Tan x = Perpendicular / Base

Tan x = 50 cm/ 18 cm = 2.778

= tan^-1 (2.778)

Trigonometry Formulas:

Trigonometric ratios of complementary angles:

First Quadrant: 

• sin(π/2−θ) = cosθ
• cos(π/2−θ) = sinθ
• tan(π/2−θ) = cotθ
• cot(π/2−θ) = tanθ
• sec(π/2−θ) = cosecθ
• cosec(π/2−θ) = secθ

Second Quadrant:

• sin(π−θ) = sinθ
• cos(π−θ) = -cosθ
• tan(π−θ) = -tanθ
• cot(π−θ) = -cotθ
• sec(π−θ) = -secθ
• cosec(π−θ) = cosecθ

Third Quadrant:

• sin(π+θ) = -sinθ
• cos(π+θ) = -cosθ
• tan(π+θ) = tanθ
• cot(π+θ) = cotθ
• sec(π+θ) = -secθ
• cosec(π+θ) = -cosecθ

Fourth Quadrant:

• sin(2π−θ) = -sinθ
• cos(2π−θ) = cosθ
• tan(2π−θ) = -tanθ
• cot(2π−θ) = -cotθ
• sec(2π−θ) = secθ
• cosec(2π−θ) = -cosecθ

Identities:

1) Periodicity Identities:

• sin(2nπ + θ) = sinθ
• cos(2nπ + θ) = cosθ
• tan(2nπ + θ) = tanθ
• cot(2nπ + θ) = cotθ
• sec(2nπ + θ) = secθ
• cosec(2nπ + θ) = cosecθ

2) Trigonometric Identities:

• sin2θ+cos2θ=1
  ⇒ sin² x = 1 – cos² x
  ⇒ sin x = √(1 – cos²x)
• tan2θ+1=sec2θ
• cot2θ+1=cosec2θ
  

Trigonometric ratios signs

• sin(−θ)=−sinθ
• cos(−θ)=cosθ
• tan(−θ)=−tanθ
• cosec(−θ)=−cosecθ
• sec(−θ)=secθ
• cot(−θ)=−cotθ

Sum and difference of two angles trigonometric ratios

• sin(A+B)=sinAcosB + cosAsinB
• sin(A−B)=sinAcosB – cosAsinB
• cos(A+B)=cosAcosB – sinAsinB
• cos(A–B)=cosAcosB + sinAsinB
• tan(A+B)=tanA+tanB1 – tanAtanB
• tan(A–B)=tanA–tanB1 + tanAtanB

3) Product Identities:

• sinA sinB=12[cos(A–B) −cos(A+B)]
• cosAcosB=12[cos(A–B) +cos(A+B)]
• sinAcosB=12[sin(A+B) +sin(A−B)]
• cosAsinB=12[sin(A+B)–sin(A−B)]

4) Sum to Product Identities:

• sinA+sinB=2sin(A+B2) cos(A−B2)
• sinA−sinB=2cos(A+B2) sin(A−B2)
• cosA+cosB=2cos(A+B2) cos(A−B2)
• cosA−cosB=–2sin(A+B2) sin(A−B2)

5) Half angle Identities:

• sinA2=±1−cosA2−−−−−−√
• cosA2=±1+cosA2−−−−−−√
• tan(A2)=1−cos(A)1+cos(A)−−−−−−√
• tan(A2)=1−cos(A)1+cos(A)−−−−−−√=(1−cos(A))(1−cos(A))(1+cos(A))(1−cos(A))−−−−−−−−−−−−−√=(1−cos(A))21−cos2(A)−−−−−−−−√=(1−cos(A))2sin2(A)−−−−−−−−√=1−cos(A)sin(A) So, tan(A2)=1−cos(A)sin(A)

6) Double angle Identities:

• sin2A=2sinAcosA=2tanA1+tan2A
• cos2A=cos2A–sin2A=1–2sin2A=2cos2A–1=1−tan2A1+tan2A
• tan2A=2tanA1–tan2A

7) Triple angle Identities:

• sin3A=3sinA–4sin3A=4sin(60∘−A).sinA.sin(60∘+A)
• cos3A=4cos3A–3cosA=4cos(60∘−A).cosA.cos(60∘+A)
• tan3A=3tanA–tan3A1−3tan2A=tan(60∘−A).tanA.tan(60∘+A)
  

Inverse Properties

• θ=sin−1(x )is equivalent to x=sinθ
• θ=cos−1(x) is equivalent to x=cosθ
• θ=tan−1(x) is equivalent to x=tanθ
• 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

Formulas:

• sin ^-1 (-x) = – sin ^-1 x
• cos ^-1 (-x) = – sin ^-1 x
• tan ^-1 (-x) = – tan ^-1 x
• cosec ^-1 (-x) = – cosec ^-1 x
• sec ^-1 (-x) = – sec ^-1 x
• cot ^-1 (-x) = – cot ^-1 x
  
• sin ^-1 (1/x) = cosec ^-1 x
• cos ^-1 (1/x) = sec ^-1 x
• tan ^-1 (1/x) = cot ^-1 x
• tan ^-1 (1/x) = cot ^-1 x
  
• sin ^-1 (x) + cos ^-1 (x) = π/2
• tan ^-1 (x) + cot ^-1 (x) = π/2
• sec ^-1 (x) + cosec ^-1 (x) = π/2

Inverse Trigonometry Substitution

Expression	Substitution	Identity
√a2 − x2	x = a sin θ	1 – sin2 θ = cos2 θ
√a2 + x2	x = a tan θ	1 – tan2 θ = sec2 θ
√x2 − a2	x = a sec θ	sec2 θ – 1 = tan2 θ

Important questions:

1. In a given triangle LMN, with a right angle at M, LN + MN = 30 cm and LM = 8 cm. Calculate the values of sin L, cos L, and tan L.
2. Find cos X and tan X if sin X = 2/3
3. Calculate general solution of the equation: tan²θ +(2 – √6) tan θ – √2 = 0
4. Prove the equation: sin^-1 (23) – sin^-1 (9/12) = cos^-1 (80/90)
5. Calculate the value of sec A if (1 + cos A) (1 – cos A) = 2/3
6. Prove that tan 3x tan 2 tan = tan 3x – tan 2 – tan
7. Calculate the value of tan^-1 a + tan^-1 b + tan^-1 c if a, b, c > 0 and a + b + c = abc.
8. In a triangle, the length of the two larger sides are 12 cm and 7 cm, respectively. If the angles of the triangle are in arithmetic progression, then find out the length of the third side in cm.
9. Calculate the value of tan X + cot Y if sin (X + Y) = 1 and tan (X – Y) = 1/√3
10. Calculate the value of sec^-1 (1/2) + 2 cosec^-1 (1/2)