Foundations of Trigonometry: Navigating Angles and Ratios with Ease"

Foundations of Trigonometry: Navigating Angles and Ratios with Ease"

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  TRIGONOMTERY Basic Trigonometry
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  CLICK HERE FORTHE FULL PLAYLIST S. No. TOPICS 1. Logarithms 2. Inequalities - Wavy Curve 3. Trigonometry 4. Functions: Basics, Domain & Range 5. Modulus & Greatest Integer Functions 6. Complex Numbers 7. Differentiation 8. Limits 9. Quadratic Equations 10. Permutations & Combinations S. No. TOPICS 11. Probability 12. Binomial Theorem 13. Integration 14. Sequence & Series 15. Vectors 16. 3D Geometry 17. Graphs & Transformation 18. Straight Lines & Circles 19. Sets & Relations 20. Conic Sections 21. Matrices and Determinants
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  TRIGONOMETRY JEE MAINS &ADVANCED COURSE ➔ Foundation Sessions for starters ➔ Complete PYQ’s (2015-2023) ➔ NTA + Cengage CHAPTER WISE Questions ➔ My HANDWRITTEN Notes tinyurl.com/jeewithnehamam WE DO NOT SELL ANY COURSES For FREE & Focused JEE MATERIAL, CLICK to Join TELEGRAM : t.me/mathematicallyinclined
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  TRIGONOMTERY ● Measurement ofan Angle ● Relation between radians and degrees ● Trigonometric Ratios ● Reciprocal and Co-ratio of Trigonometric Ratios ● Sign of Trigonometric Functions ● Quadrant Angles and Allied Angles ● 3- STEPS to get any ANGLE ● Trigonometric functions of particular angles ● Trigonometric Identities ● Trigonometric Ratios of Negative Angles ● Compound Angles ● Double Angle & Triple Angle Formula TRIGONOMETRY
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  TRIGONOMTERY JEE(MAIN) PYQs UsingTRIGONOMETRY BASICS
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  TRIGONOMTERY JEE(MAIN) PYQs UsingTRIGONOMETRY BASICS
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  TRIGONOMTERY Angle Measurements The amountof rotation of a ‘moving ray' (terminating ray) with reference to a fixed ray' (initial ray) is called an angle. And it is denoted by θ or α or β etc. θ
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  TRIGONOMTERY Positive Angle If therotation of the terminating ray is in anti-clockwise direction, the angle is called as positive. Negative Angle If the rotation of the terminating ray is in clockwise direction, the angle is called as negative.
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  TRIGONOMTERY System of Measurementof an Angle (i) The Sexagesimal Measurement (ii) The circular Measurement
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  TRIGONOMTERY Definition 1. Sexagesimal system(British System) It is denoted by 10 If the central angle is divided into 360 equal parts, each part in it is called One degree.
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  TRIGONOMTERY Minute 10 is dividedinto 60 equal parts each part in it called One minute. It is denoted as 1′. 10 = 60′ Second Again if 1’ is divided into 60 equal parts each part in it called One second. It is denoted as 1’’. 1’ = 60’’
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  TRIGONOMTERY Important Points
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  TRIGONOMTERY Definition A radian isan angle subtended at the centre of a circle by an arc whose length is equal to the radius of the circle. r B A r r 1c 2. Circular system (or) Radian Measure
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  TRIGONOMTERY One radian isdenoted as 1c. Definition Angle subtended at the centre of a circle of radius r by an arc of length l is defined as radians. l r θ =
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  TRIGONOMTERY Relation between radiansand degrees ∴ 1c = 1800 π 1c ≈ (57.272…)0 3600 = 2πc 1c = (180)0 22 7 = 0 11 630 90 11
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  TRIGONOMTERY Relation between radiansand degrees Remember ➢ To convert radians into degree multiply with ➢ To convert degrees into radians multiply with π 1800 1800 π
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  TRIGONOMTERY Q. Express thefollowing angles in degrees.
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  TRIGONOMTERY Solution:
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  TRIGONOMTERY Q. Express thefollowing angles in degrees. 2 9 π
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  TRIGONOMTERY Solution:
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  TRIGONOMTERY Q. Express thefollowing angles in degrees.
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  TRIGONOMTERY Solution:
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  TRIGONOMTERY Q. Determine howthe angle radians are classified. A B C D Acute Right Obtuse Reflex 7 8 π
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  TRIGONOMTERY Solution:
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  TRIGONOMTERY A B C D Acute Right Obtuse Reflex Q. Determine howthe angle radians are classified. 7 8 π
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  TRIGONOMTERY Q. Convert eachdegree measure to radian measure. a. 135° b. 40°
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  TRIGONOMTERY
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  TRIGONOMTERY Q. Convert 7°30'into radians
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  TRIGONOMTERY
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  TRIGONOMTERY Trigonometric Ratios ● Sinθ= Opposite side Hypotenuse ● Cosθ = Adjacent side Hypotenuse ● Tanθ = Opposite side Adjacent side ● Cosecθ = Hypotenuse Opposite side ● Secθ = Hypotenuse Adjacent side ● Cotθ = Adjacent side Opposite side
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  TRIGONOMTERY Reciprocal and Co-ratioof Trigonometric Ratios
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  TRIGONOMTERY
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  TRIGONOMTERY TRIGONOMETRIC IDENTITIES 1. sin2θ+ cos2θ = 1 2. sec2θ – tan2θ = 1 3. cosec2θ – cot2θ = 1
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  TRIGONOMTERY Values Of T-RatiosAt Particular Angles
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  TRIGONOMTERY sinθ cosθ tanθ 00 0 1 0 1 2 1 √2 2 √3 1 √2 1 √3 1 300 450 1 2 √3 0 Undefined 900 2 √3 600 1 0 π 6 π 4 π 3 π 2 T-ratio Angle(θ) Trigonometric functions of particular angles
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  TRIGONOMTERY tanθ 0 0 T-ratio Angle(θ) sinθ cosθ 1800 3600 π 2π –1 0 1 0 Trigonometric functions of particular angles
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  TRIGONOMTERY Q.
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  TRIGONOMTERY
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  TRIGONOMTERY Sign of TrigonometricFunctions I II III IV Y X
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  TRIGONOMTERY cosθ < 0; sinθ < 0 cosθ < 0 ; sinθ > 0 cosθ > 0 ; sinθ > 0 cosθ > 0 ; sinθ < 0 I II III IV x y x > 0, y > 0 x < 0, y > 0 x < 0, y < 0 x > 0, y < 0 Sign of Trigonometric Functions
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  TRIGONOMTERY Trigonometric Ratios ofNegative Angles sin(-θ) -sinθ cos(-θ) cosθ tan(-θ) -tanθ cot(-θ) -cotθ sec(-θ) secθ cosec(-θ) -cosecθ
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  TRIGONOMTERY Quadrant Angles andAllied Angles
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  TRIGONOMTERY STEPS to getany ANGLE
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  TRIGONOMTERY Q. Find thevalue of the following trigonometric ratios. (i) sin(1500) (ii) cos(-2100)
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  TRIGONOMTERY
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  TRIGONOMTERY Trigonometric Ratios OfCompound Angles
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  TRIGONOMTERY Sine And CosineOf Compound Angles Sum of any two or more angles is called compound angle. 1) sin (α + β) 3) cos (α + β) sinα cosβ + cosα sinβ = = cosα cosβ – sin α sin β sin (α – β) 2) = sinα cosβ – cosα sinβ cos (α – β) = cos α cos β + sin α sin β. 4)
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  TRIGONOMTERY Sine And CosineOf Compound Angles 3) cos (α + β) = cosα cosβ – sin α sin β sin (α – β) 2) = sinα cosβ – cosα sinβ cos (α – β) = cos α cos β + sin α sin β. 4)
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  TRIGONOMTERY tan (α +β) = tan α + tan β 1 – tan α tan β tan (α – β) = tan α – tan β 1 + tan α tan β = sin (α + β) cos (α + β) = sin (α – β) cos (α – β) Similarly,
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  TRIGONOMTERY Q. Find thevalue of sin 150
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  TRIGONOMTERY sin 150 = = √3 2 – 1 √2 1 2 = √3 –1 2√2 1 √2 sin (450 – 300) = sin 450 cos300 . – cos450 . sin300 Solution:
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  TRIGONOMTERY Q. Find thevalue of cos⁡ 75∘.
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  TRIGONOMTERY
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  TRIGONOMTERY A B D C -1 -2 1 2 Q. The Valueof is
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  TRIGONOMTERY Solution:
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  TRIGONOMTERY A B D C -1 -2 1 2 Q. The Valueof is
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  TRIGONOMTERY Double Angle
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  TRIGONOMTERY Trigonometric functions ofDouble Angles I. sin 2θ = 2 sin θ . cos θ Proof:
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  TRIGONOMTERY I. sin 2θ= 2 sin θ . cos θ sin 2θ = sin (θ + θ) = sin θ . cos θ + cos θ . sin θ = 2 sin θ . cos θ Trigonometric functions of Double Angles
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  TRIGONOMTERY II. cos 2θ= cos2 θ – sin2 θ = 1 – 2 sin2 θ = 2 cos2 θ – 1 Proof: Trigonometric functions of Double Angles
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  TRIGONOMTERY II. cos 2θ= cos2 θ – sin2 θ = 1 – 2 sin2 θ = 2 cos2 θ – 1 cos 2θ = cos (θ + θ) = cos θ . cos θ – sin θ . sin θ ∴ cos 2θ = cos2 θ – sin2 θ …. (i) = (1 – sin2 θ) – sin2 θ …. (ii) = 1 – 2 sin2 θ Also, cos 2θ = cos2 θ – sin2 θ = cos2 θ – (1 – cos2 θ) = cos2 θ – 1 + cos2 θ = 2 cos2 θ – 1 …. (iii) Trigonometric functions of Double Angles
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  TRIGONOMTERY Q. Given equationis equivalent to A B D C cos 2x sin 2x sin x cos x
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  TRIGONOMTERY Solution:
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  TRIGONOMTERY Q. Given equationis equivalent to A B D C cos 2x sin 2x sin x cos x
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  TRIGONOMTERY III. tan 2θ= 2 tan θ 1 – tan2 θ Proof: Trigonometric functions of Double Angles
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  TRIGONOMTERY III. tan 2θ tan 2θ= 2 tan θ 1 – tan2 θ = tan (θ + θ) = tan θ + tan θ 1 – tan θ . tan θ ∴ tan 2θ = 2 tan θ 1 – tan2 θ Trigonometric functions of Double Angles
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  TRIGONOMTERY Q.Prove: 2 tanx 1 + tan2 x i) sin 2x = ii) cos 2x = 1– tan2 x 1 + tan2 x
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  TRIGONOMTERY i) R.H.S= 2 sinx / cos x 1 + sin2 x cos2 x 2 tan x 1 + tan2 x = = 2 sin x cos x × (cos2 x + sin2 x) cos2 x 2 sin x cos x = sin 2x = L.H.S = Solution:
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  TRIGONOMTERY ii) R.H.S = 1–tan2 x 1 + tan2 x 1 – sin2 x cos2 x 1 + sin2 x cos2 x = = cos2 x − sin2 x cos2 x cos2 x + sin2 x cos2 x = cos2 x – sin2 x cos2 x + sin2 x = cos 2x = L.H.S Solution:
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  TRIGONOMTERY Q.
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  TRIGONOMTERY
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  TRIGONOMTERY Triple Angle Formulae
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  TRIGONOMTERY Triple Angle Formulae ● ● ●
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  TRIGONOMTERY Q. If sinA=3/4 then value of sin3A will be: A B D C 9/16 -9/16 9/32 7/16
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  TRIGONOMTERY
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  TRIGONOMTERY Q. If sinA=3/4 then value of sin3A will be: A B D C 9/16 -9/16 9/32 7/16
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  TRIGONOMTERY C and DFormulae
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  TRIGONOMTERY Q. Express sin6θ– sin2θ as a product.
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  TRIGONOMTERY
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  TRIGONOMTERY Defactorization Formulae (i) 2sinAcosB= sin(A+B) + sin(A–B) (ii) 2cosAsinB = sin(A+B) – sin(A–B) (iii) 2cosAcosB = cos(A+B) + cos(A–B) (iv) –2sinAsinB = cos(A+B) – cos(A–B)
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  TRIGONOMTERY Q. Express 2Cos7x Cos3x as a Sum.
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  TRIGONOMTERY