Remember that 10th 12th maths subject bring the topic trigonometry and its formula? The education of side length and angles of triangle and its relation is 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.
Trigonometry formulas are very helpful in solving problems in a better way. The students should have great trigonometry skills that allow them to work out complex angles and dimensions. The main objective of trigonometry is to measure distances in an accurate way. The students should know about algebra and geometry before learning trigonometry.
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. Trigonometry is also used in the fields of astronomy, engineering, music production, computer imaging, medical imaging, biology, chemistry and a lot more! This shows the importance of studying trigonometry, even after school.
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. If the students thoroughly understand trigonometry at the introductory stage, they won’t face much difficulty in comprehending the advanced concepts too.
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.
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)
This is all the important information related to the maths topic trigonometry. We hope you found the article informative as we have covered all the mandatory data related to trigonometry. IF you want to read such more informative educational article then do follow study woo thank you.
