🔹 1. Introduction

Trigonometric Functions relate angles to ratios of sides of a triangle and are fundamental in mathematics.

👉 Used in:

• Geometry
• Calculus
• Physics (waves, motion)
• Engineering

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🔹 2. Angles and Measurement

✔️ Degree Measure

• Full circle = 360°

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✔️ Radian Measure

• Full circle = 2π radians

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✔️ Conversion

$180^\circ = \pi \text{ radians}$180∘=π radians

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🔹 3. Trigonometric Functions

For angle θ:

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✔️ Basic Functions

$\sin\theta = \frac{\text{Perpendicular}}{\text{Hypotenuse}}$sinθ=HypotenusePerpendicular​

$\cos\theta = \frac{\text{Base}}{\text{Hypotenuse}}$cosθ=HypotenuseBase​

$\tan\theta = \frac{\text{Perpendicular}}{\text{Base}}$tanθ=BasePerpendicular​

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✔️ Reciprocal Functions

$\csc\theta = \frac{1}{\sin\theta}$cscθ=sinθ1​

$\sec\theta = \frac{1}{\cos\theta}$secθ=cosθ1​

$\cot\theta = \frac{1}{\tan\theta}$cotθ=tanθ1​

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🔹 4. Trigonometric Identities

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✔️ Fundamental Identities

$\sin^2\theta + \cos^2\theta = 1$sin2θ+cos2θ=1

$\theta$θ

$\sin^2\theta \approx 0.329,\;\cos^2\theta \approx 0.671$sin2θ≈0.329,cos2θ≈0.671

$\sin^2\theta + \cos^2\theta \approx 1$sin2θ+cos2θ≈1θ = 35°|cos θ| = 0.819|sin θ| = 0.574cos² θsin² θ0.671 + 0.329 = 1

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$1 + \tan^2\theta = \sec^2\theta$1+tan2θ=sec2θ

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$1 + \cot^2\theta = \csc^2\theta$1+cot2θ=csc2θ

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🔹 5. Values of Trigonometric Functions

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✔️ Standard Angles

θ	sinθ	cosθ	tanθ
0°	0	1	0
30°	1/2	√3/2	1/√3
45°	1/√2	1/√2	1
60°	√3/2	1/2	√3
90°	1	0	∞

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🔹 6. Signs of Functions (Quadrants)

Using ASTC rule:

• 1st → All positive
• 2nd → sin positive
• 3rd → tan positive
• 4th → cos positive

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🔹 7. Trigonometric Functions of Sum and Difference

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✔️ Sine

$\sin(A+B) = \sin A \cos B + \cos A \sin B$sin(A+B)=sinAcosB+cosAsinB

$a$a

$b$b

$\sin\left(a+b\right) \approx 0.866$sin(a+b)≈0.866

$\sin\left(a\right)\cos\left(b\right) + \cos\left(a\right)\sin\left(b\right) \approx 0.866$sin(a)cos(b)+cos(a)sin(b)≈0.866aba+b

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✔️ Cosine

$\cos(A+B) = \cos A \cos B – \sin A \sin B$cos(A+B)=cosAcosB−sinAsinB

$a$a

$b$b

$\cos\left(a+b\right) \approx 0.5$cos(a+b)≈0.5

$\cos\left(a\right)\cos\left(b\right) – \sin\left(a\right)\sin\left(b\right) \approx 0.5$cos(a)cos(b)−sin(a)sin(b)≈0.5aba+b

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✔️ Tangent

$\tan(A+B) = \frac{\tan A + \tan B}{1 – \tan A \tan B}$tan(A+B)=1−tanAtanBtanA+tanB​

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🔹 8. Double Angle Formulas

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✔️ Sine

$\sin 2A = 2\sin A \cos A$sin2A=2sinAcosA

$a$a

$a$a

$\sin\left(a+a\right) \approx 0.866$sin(a+a)≈0.866

$\sin\left(a\right)\cos\left(a\right) + \cos\left(a\right)\sin\left(a\right) \approx 0.866$sin(a)cos(a)+cos(a)sin(a)≈0.866aaa+a

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✔️ Cosine

$\cos 2A = \cos^2 A – \sin^2 A$cos2A=cos2A−sin2A

$a$a

$a$a

$\cos\left(a+a\right) \approx 0.5$cos(a+a)≈0.5

$\cos\left(a\right)\cos\left(a\right) – \sin\left(a\right)\sin\left(a\right) \approx 0.5$cos(a)cos(a)−sin(a)sin(a)≈0.5aaa+a

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✔️ Tangent

$\tan 2A = \frac{2\tan A}{1 – \tan^2 A}$tan2A=1−tan2A2tanA​

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🔹 9. Trigonometric Functions of Multiple Angles

✔️ sin 3A, cos 3A (advanced)

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🔹 10. General Solution of Trigonometric Equations

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✔️ sin x = sin α

$x = n\pi + (-1)^n \alpha$x=nπ+(−1)nα

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✔️ cos x = cos α

$x = 2n\pi \pm \alpha$x=2nπ±α

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✔️ tan x = tan α

$x = n\pi + \alpha$x=nπ+α

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🔹 11. Graphs of Trigonometric Functions (Concept)

• sin x → wave pattern
• cos x → wave shifted
• tan x → repeating curve

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🔹 12. Important Results

✔️ sin²θ + cos²θ = 1
✔️ tanθ = sinθ / cosθ
✔️ Periodicity:

• sin x, cos x → 2π
• tan x → π

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🔹 13. Applications

✔️ Wave motion
✔️ Sound & light
✔️ Engineering design
✔️ Navigation

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🔹 14. JEE & CBSE Important Points

✔️ Identities are very important
✔️ Learn standard values
✔️ Practice equation solving
✔️ Focus on transformations
✔️ Angle conversion frequently asked

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🔹 15. Common Mistakes

❌ Sign errors in quadrants
❌ Wrong identities
❌ Confusing radians and degrees
❌ Calculation mistakes

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🔹 16. Practice Questions

1. Prove identities
2. Evaluate trig expressions
3. Solve trig equations
4. Find general solutions
5. Simplify expressions

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🔹 17. Quick Revision Sheet

• sin²θ + cos²θ = 1
• sin(A+B), cos(A+B)
• sin2A = 2sinA cosA
• tan2A = 2tanA/(1−tan²A)
• 180° = π radians

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🔹 18. Conclusion

Trigonometric Functions is a core and high-weightage chapter in Class 11.

👉 Strong understanding helps in:

• Calculus
• Coordinate Geometry
• JEE problem solving