6.1 Trigonometry

Detailed Theory: Trigonometry

1. Basic Concepts and Angles

1.1 What is Trigonometry?

Trigonometry is the study of relationships between angles and sides of triangles.

Etymology: Greek words "trigonon" (triangle) + "metron" (measure)

1.2 Angle Measurement Systems

a) Degree Measure

• Full circle = $360^\circ$

• Right angle = $90^\circ$

• Straight angle = $180^\circ$

b) Radian Measure

• More natural for mathematics

• Based on arc length

• Full circle = $2\pi$ radians

• $180^\circ = \pi$ radians

c) Conversion Formulas

$$\text{Degrees to Radians: } \theta_{\text{rad}} = \frac{\pi}{180} \times \theta_{\text{deg}}$$

$$\text{Radians to Degrees: } \theta_{\text{deg}} = \frac{180}{\pi} \times \theta_{\text{rad}}$$

d) Common Conversions

$$30^\circ = \frac{\pi}{6} \text{ radians}$$

$$45^\circ = \frac{\pi}{4} \text{ radians}$$

$$60^\circ = \frac{\pi}{3} \text{ radians}$$

$$90^\circ = \frac{\pi}{2} \text{ radians}$$

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

$$360^\circ = 2\pi \text{ radians}$$

1.3 Types of Angles

1. Acute angle: $0^\circ < \theta < 90^\circ$

2. Right angle: $\theta = 90^\circ$

3. Obtuse angle: $90^\circ < \theta < 180^\circ$

4. Straight angle: $\theta = 180^\circ$

5. Reflex angle: $180^\circ < \theta < 360^\circ$

6. Complete angle: $\theta = 360^\circ$

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2. Trigonometric Ratios

2.1 Right Triangle Trigonometry

Consider right triangle with:

• Hypotenuse (opposite right angle)

• Opposite side (opposite to angle $\theta$)

• Adjacent side (next to angle $\theta$)

a) Basic Ratios

1. Sine: $\sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$

2. Cosine: $\cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}$

3. Tangent: $\tan\theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{\sin\theta}{\cos\theta}$

b) Reciprocal Ratios

1. Cosecant: $\csc\theta = \frac{1}{\sin\theta} = \frac{\text{Hypotenuse}}{\text{Opposite}}$

2. Secant: $\sec\theta = \frac{1}{\cos\theta} = \frac{\text{Hypotenuse}}{\text{Adjacent}}$

3. Cotangent: $\cot\theta = \frac{1}{\tan\theta} = \frac{\text{Adjacent}}{\text{Opposite}} = \frac{\cos\theta}{\sin\theta}$

2.2 Mnemonic for Remembering

SOH-CAH-TOA:

• Sine = Opposite/Hypotenuse

• Cosine = Adjacent/Hypotenuse

• Tangent = Opposite/Adjacent

2.3 Values for Standard Angles

Angle = $30^\circ$ or $\frac{\pi}{6}$

$$\sin 30^\circ = \frac{1}{2}$$

$$\cos 30^\circ = \frac{\sqrt{3}}{2}$$

$$\tan 30^\circ = \frac{1}{\sqrt{3}}$$

Angle = $45^\circ$ or $\frac{\pi}{4}$

$$\sin 45^\circ = \frac{1}{\sqrt{2}}$$

$$\cos 45^\circ = \frac{1}{\sqrt{2}}$$

$$\tan 45^\circ = 1$$

Angle = $60^\circ$ or $\frac{\pi}{3}$

$$\sin 60^\circ = \frac{\sqrt{3}}{2}$$

$$\cos 60^\circ = \frac{1}{2}$$

$$\tan 60^\circ = \sqrt{3}$$

2.4 Trigonometric Table (0° to 90°)

Angle	$\sin$	$\cos$	$\tan$
$0^\circ$	0	1	0
$30^\circ$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{3}}$
$45^\circ$	$\frac{1}{\sqrt{2}}$	$\frac{1}{\sqrt{2}}$	1
$60^\circ$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$
$90^\circ$	1	0	$\infty$

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3. Trigonometric Identities

3.1 Pythagorean Identities

1. $\sin^2\theta + \cos^2\theta = 1$

2. $1 + \tan^2\theta = \sec^2\theta$

3. $1 + \cot^2\theta = \csc^2\theta$

3.2 Reciprocal Identities

1. $\csc\theta = \frac{1}{\sin\theta}$

2. $\sec\theta = \frac{1}{\cos\theta}$

3. $\cot\theta = \frac{1}{\tan\theta}$

3.3 Quotient Identities

1. $\tan\theta = \frac{\sin\theta}{\cos\theta}$

2. $\cot\theta = \frac{\cos\theta}{\sin\theta}$

3.4 Co-function Identities

1. $\sin(90^\circ - \theta) = \cos\theta$

2. $\cos(90^\circ - \theta) = \sin\theta$

3. $\tan(90^\circ - \theta) = \cot\theta$

4. $\cot(90^\circ - \theta) = \tan\theta$

5. $\sec(90^\circ - \theta) = \csc\theta$

6. $\csc(90^\circ - \theta) = \sec\theta$

3.5 Even-Odd Identities

1. $\sin(-\theta) = -\sin\theta$ (odd function)

2. $\cos(-\theta) = \cos\theta$ (even function)

3. $\tan(-\theta) = -\tan\theta$ (odd function)

3.6 Periodicity Identities

1. $\sin(\theta + 2\pi) = \sin\theta$

2. $\cos(\theta + 2\pi) = \cos\theta$

3. $\tan(\theta + \pi) = \tan\theta$

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4. Trigonometric Functions of Any Angle

4.1 Unit Circle Approach

Unit circle: Circle with radius 1 centered at origin

For angle $\theta$ measured counterclockwise from positive x-axis:

• Point on circle: $(\cos\theta, \sin\theta)$

• x-coordinate = $\cos\theta$

• y-coordinate = $\sin\theta$

4.2 Signs in Different Quadrants

Quadrant	$\sin$	$\cos$	$\tan$
I (0°-90°)	+	+	+
II (90°-180°)	+	-	-
III (180°-270°)	-	-	+
IV (270°-360°)	-	+	-

Mnemonic: "All Students Take Calculus"

• All: All positive in Quadrant I

• Students: Sine positive in Quadrant II

• Take: Tangent positive in Quadrant III

• Calculus: Cosine positive in Quadrant IV

4.3 Reference Angles

Reference angle = Acute angle between terminal side and x-axis

Finding reference angle $\alpha$:

• Quadrant I: $\alpha = \theta$

• Quadrant II: $\alpha = 180^\circ - \theta$

• Quadrant III: $\alpha = \theta - 180^\circ$

• Quadrant IV: $\alpha = 360^\circ - \theta$

Using reference angle:

$$\sin\theta = \pm\sin\alpha$$

$$\cos\theta = \pm\cos\alpha$$

$$\tan\theta = \pm\tan\alpha$$

Sign depends on quadrant.

4.4 Example: Find $\sin 210^\circ$

Step 1: $210^\circ$ is in Quadrant III (180°-270°)

Step 2: Reference angle = $210^\circ - 180^\circ = 30^\circ$

Step 3: In Quadrant III, sine is negative

Step 4: $\sin 210^\circ = -\sin 30^\circ = -\frac{1}{2}$

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5. Sum and Difference Formulas

5.1 Sine Formulas

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

$$\sin(A - B) = \sin A \cos B - \cos A \sin B$$

5.2 Cosine Formulas

$$\cos(A + B) = \cos A \cos B - \sin A \sin B$$

$$\cos(A - B) = \cos A \cos B + \sin A \sin B$$

5.3 Tangent Formulas

$$\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$$

$$\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$$

5.4 Double Angle Formulas

$$\sin 2A = 2 \sin A \cos A$$

$$\cos 2A = \cos^2 A - \sin^2 A = 2\cos^2 A - 1 = 1 - 2\sin^2 A$$

$$\tan 2A = \frac{2\tan A}{1 - \tan^2 A}$$

5.5 Half Angle Formulas

$$\sin\frac{A}{2} = \pm\sqrt{\frac{1 - \cos A}{2}}$$

$$\cos\frac{A}{2} = \pm\sqrt{\frac{1 + \cos A}{2}}$$

$$\tan\frac{A}{2} = \pm\sqrt{\frac{1 - \cos A}{1 + \cos A}} = \frac{1 - \cos A}{\sin A} = \frac{\sin A}{1 + \cos A}$$

5.6 Triple Angle Formulas

$$\sin 3A = 3\sin A - 4\sin^3 A$$

$$\cos 3A = 4\cos^3 A - 3\cos A$$

$$\tan 3A = \frac{3\tan A - \tan^3 A}{1 - 3\tan^2 A}$$

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6. Product-to-Sum and Sum-to-Product Formulas

6.1 Product-to-Sum Formulas

$$\sin A \cos B = \frac{1}{2}[\sin(A+B) + \sin(A-B)]$$

$$\cos A \sin B = \frac{1}{2}[\sin(A+B) - \sin(A-B)]$$

$$\cos A \cos B = \frac{1}{2}[\cos(A+B) + \cos(A-B)]$$

$$\sin A \sin B = -\frac{1}{2}[\cos(A+B) - \cos(A-B)]$$

6.2 Sum-to-Product Formulas

$$\sin A + \sin B = 2\sin\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right)$$

$$\sin A - \sin B = 2\cos\left(\frac{A+B}{2}\right)\sin\left(\frac{A-B}{2}\right)$$

$$\cos A + \cos B = 2\cos\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right)$$

$$\cos A - \cos B = -2\sin\left(\frac{A+B}{2}\right)\sin\left(\frac{A-B}{2}\right)$$

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7. Trigonometric Equations

7.1 Basic Solution Method

To solve $\sin\theta = k$ or $\cos\theta = k$ or $\tan\theta = k$:

1. Find principal solution (using inverse functions)

2. Use periodicity to find general solution

7.2 General Solutions

a) For $\sin\theta = \sin\alpha$

$$\theta = n\pi + (-1)^n\alpha, \quad n \in \mathbb{Z}$$

b) For $\cos\theta = \cos\alpha$

$$\theta = 2n\pi \pm \alpha, \quad n \in \mathbb{Z}$$

c) For $\tan\theta = \tan\alpha$

$$\theta = n\pi + \alpha, \quad n \in \mathbb{Z}$$

7.3 Examples

Example 1: Solve $\sin\theta = \frac{1}{2}$

Solution:

Reference angle: $\sin 30^\circ = \frac{1}{2}$

General solution:

$$\theta = n\pi + (-1)^n \frac{\pi}{6}, \quad n \in \mathbb{Z}$$

Specific solutions in $[0, 2\pi)$: $\frac{\pi}{6}$, $\frac{5\pi}{6}$

Example 2: Solve $\cos\theta = -\frac{1}{2}$

Solution:

Reference angle: $\cos 60^\circ = \frac{1}{2}$

Since cosine is negative in Quadrants II and III:

In Quadrant II: $\theta = 180^\circ - 60^\circ = 120^\circ$ or $\frac{2\pi}{3}$

In Quadrant III: $\theta = 180^\circ + 60^\circ = 240^\circ$ or $\frac{4\pi}{3}$

General solution: $\theta = 2n\pi \pm \frac{2\pi}{3}$

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8. Graphs of Trigonometric Functions

8.1 Sine Function: $y = \sin x$

• Domain: All real numbers

• Range: $[-1, 1]$

• Period: $2\pi$

• Amplitude: 1

• Zeros: $x = n\pi$, $n \in \mathbb{Z}$

• Maximum: $1$ at $x = \frac{\pi}{2} + 2n\pi$

• Minimum: $-1$ at $x = \frac{3\pi}{2} + 2n\pi$

8.2 Cosine Function: $y = \cos x$

• Domain: All real numbers

• Range: $[-1, 1]$

• Period: $2\pi$

• Amplitude: 1

• Zeros: $x = \frac{\pi}{2} + n\pi$

• Maximum: $1$ at $x = 2n\pi$

• Minimum: $-1$ at $x = \pi + 2n\pi$

8.3 Tangent Function: $y = \tan x$

• Domain: $x \neq \frac{\pi}{2} + n\pi$

• Range: All real numbers

• Period: $\pi$

• Vertical asymptotes: $x = \frac{\pi}{2} + n\pi$

• Zeros: $x = n\pi$

8.4 General Form: $y = A\sin(Bx + C) + D$

• A: Amplitude (vertical stretch)

• B: Affects period (Period = $\frac{2\pi}{|B|}$)

• C: Phase shift (horizontal shift = $-\frac{C}{B}$)

• D: Vertical shift

8.5 Example: Graph $y = 2\sin(3x - \pi) + 1$

Amplitude: $2$

Period: $\frac{2\pi}{3}$

Phase shift: $\frac{\pi}{3}$ to the right

Vertical shift: $1$ up

Range: $[-1, 3]$

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9. Inverse Trigonometric Functions

9.1 Definitions and Ranges

a) Inverse Sine: $y = \sin^{-1}x$ or $y = \arcsin x$

• Domain: $[-1, 1]$

• Range: $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$

b) Inverse Cosine: $y = \cos^{-1}x$ or $y = \arccos x$

• Domain: $[-1, 1]$

• Range: $[0, \pi]$

c) Inverse Tangent: $y = \tan^{-1}x$ or $y = \arctan x$

• Domain: All real numbers

• Range: $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$

9.2 Important Properties

1. $\sin^{-1}(\sin x) = x$ only if $x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$

2. $\sin(\sin^{-1} x) = x$ for $x \in [-1, 1]$

3. $\cos^{-1}(-x) = \pi - \cos^{-1}x$

4. $\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2}$

5. $\tan^{-1}x + \cot^{-1}x = \frac{\pi}{2}$

6. $\sec^{-1}x + \csc^{-1}x = \frac{\pi}{2}$

9.3 Example: Find $\sin^{-1}\left(\frac{1}{2}\right)$

We need angle in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ with sine = $\frac{1}{2}$

$$\sin^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{6}$$

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10. Applications of Trigonometry

10.1 Solving Triangles

a) Law of Sines

For any triangle with sides $a, b, c$ opposite angles $A, B, C$:

$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$$

where $R$ is circumradius.

b) Law of Cosines

$$a^2 = b^2 + c^2 - 2bc\cos A$$

$$b^2 = a^2 + c^2 - 2ac\cos B$$

$$c^2 = a^2 + b^2 - 2ab\cos C$$

c) Law of Tangents

$$\frac{a-b}{a+b} = \frac{\tan\left(\frac{A-B}{2}\right)}{\tan\left(\frac{A+B}{2}\right)}$$

10.2 Area Formulas

1. Using base and height: $A = \frac{1}{2}bh$

2. Using two sides and included angle: $A = \frac{1}{2}ab\sin C$

3. Heron's formula: $A = \sqrt{s(s-a)(s-b)(s-c)}$ where $s = \frac{a+b+c}{2}$

10.3 Example: Solve triangle with $a=5$, $b=6$, $C=60^\circ$

Step 1: Find side $c$ using Law of Cosines:

$$c^2 = a^2 + b^2 - 2ab\cos C = 25 + 36 - 2(5)(6)\cos 60^\circ$$

$$c^2 = 61 - 60 \times \frac{1}{2} = 61 - 30 = 31$$

$$c = \sqrt{31} \approx 5.57$$

Step 2: Find angle $A$ using Law of Sines:

$$\frac{\sin A}{a} = \frac{\sin C}{c}$$

$$\sin A = \frac{a\sin C}{c} = \frac{5 \times \sin 60^\circ}{\sqrt{31}} = \frac{5 \times \frac{\sqrt{3}}{2}}{\sqrt{31}} = \frac{5\sqrt{3}}{2\sqrt{31}}$$

$$A = \sin^{-1}\left(\frac{5\sqrt{3}}{2\sqrt{31}}\right)$$

Step 3: Find angle $B$:

$$B = 180^\circ - A - C$$

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11. Trigonometric Series and Complex Numbers

11.1 Euler's Formula

$$e^{i\theta} = \cos\theta + i\sin\theta$$

11.2 De Moivre's Theorem

For any integer $n$:

$$(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)$$

11.3 Trigonometric Form of Complex Number

For complex number $z = x + iy$:

$$z = r(\cos\theta + i\sin\theta) = re^{i\theta}$$

where $r = \sqrt{x^2 + y^2}$ and $\theta = \tan^{-1}\left(\frac{y}{x}\right)$

11.4 Applications

1. Finding $n$-th roots: $n$-th roots of $z = r(\cos\theta + i\sin\theta)$ are:

   $$z_k = r^{1/n}\left[\cos\left(\frac{\theta + 2k\pi}{n}\right) + i\sin\left(\frac{\theta + 2k\pi}{n}\right)\right]$$

   for $k = 0, 1, \ldots, n-1$

2. Expressing $\cos^n\theta$ and $\sin^n\theta$ in terms of multiple angles

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12. Trigonometric Inequalities

12.1 Solving Basic Inequalities

Example: Solve $\sin x > \frac{1}{2}$ for $x \in [0, 2\pi)$

Solution:

From unit circle: $\sin x = \frac{1}{2}$ at $x = \frac{\pi}{6}$ and $x = \frac{5\pi}{6}$

Since sine is positive in Quadrants I and II:

Solution: $\frac{\pi}{6} < x < \frac{5\pi}{6}$

12.2 General Method

1. Solve corresponding equation

2. Identify intervals where inequality holds

3. Consider periodic nature

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13. Important Formulas Summary

13.1 Pythagorean Identities

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

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

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

13.2 Sum and Difference Formulas

$$\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$$

$$\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$$

13.3 Double Angle Formulas

$$\sin 2A = 2\sin A \cos A$$

$$\cos 2A = \cos^2 A - \sin^2 A$$

13.4 Half Angle Formulas

$$\sin\frac{A}{2} = \pm\sqrt{\frac{1 - \cos A}{2}}$$

$$\cos\frac{A}{2} = \pm\sqrt{\frac{1 + \cos A}{2}}$$

13.5 Law of Sines and Cosines

$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$

$$a^2 = b^2 + c^2 - 2bc\cos A$$

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14. Solved Examples

Example 1: Simplify $\frac{\sin^2\theta - \cos^2\theta}{\sin\theta \cos\theta}$

Solution:

Using identities: $\sin^2\theta - \cos^2\theta = -\cos 2\theta$

Also: $\sin\theta\cos\theta = \frac{1}{2}\sin 2\theta$

So:

$$\frac{\sin^2\theta - \cos^2\theta}{\sin\theta \cos\theta} = \frac{-\cos 2\theta}{\frac{1}{2}\sin 2\theta} = -2\cot 2\theta$$

Example 2: Prove $\frac{1 + \sin\theta}{\cos\theta} = \frac{\cos\theta}{1 - \sin\theta}$

Solution:

Cross multiply: $(1 + \sin\theta)(1 - \sin\theta) = \cos^2\theta$

Left side: $1 - \sin^2\theta = \cos^2\theta$ (using $\sin^2\theta + \cos^2\theta = 1$)

Right side: $\cos^2\theta$

Both sides equal, identity proved.

Example 3: Solve $2\cos^2x - 3\cos x + 1 = 0$ for $0 \leq x < 2\pi$

Solution:

Let $t = \cos x$, then $2t^2 - 3t + 1 = 0$

Factor: $(2t - 1)(t - 1) = 0$

So $t = \frac{1}{2}$ or $t = 1$

Case 1: $\cos x = \frac{1}{2}$ Solutions: $x = \frac{\pi}{3}$, $x = \frac{5\pi}{3}$

Case 2: $\cos x = 1$ Solution: $x = 0$

Total solutions: $x = 0$, $\frac{\pi}{3}$, $\frac{5\pi}{3}$

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15. Exam Tips and Common Mistakes

15.1 Common Mistakes

1. Confusing degrees and radians: Always check which is being used

2. Forgetting domain restrictions for inverse functions

3. Sign errors when using reference angles

4. Not considering all solutions in trigonometric equations

5. Misapplying identities (especially with signs)

15.2 Problem-Solving Strategy

1. Identify knowns and unknowns

2. Choose appropriate identities/formulas

3. Simplify step by step

4. Check domain/range restrictions

5. Verify solution when possible

15.3 Important Values to Memorize

• $\sin 0^\circ = 0$, $\cos 0^\circ = 1$, $\tan 0^\circ = 0$

• $\sin 30^\circ = \frac{1}{2}$, $\cos 30^\circ = \frac{\sqrt{3}}{2}$, $\tan 30^\circ = \frac{1}{\sqrt{3}}$

• $\sin 45^\circ = \frac{1}{\sqrt{2}}$, $\cos 45^\circ = \frac{1}{\sqrt{2}}$, $\tan 45^\circ = 1$

• $\sin 60^\circ = \frac{\sqrt{3}}{2}$, $\cos 60^\circ = \frac{1}{2}$, $\tan 60^\circ = \sqrt{3}$

• $\sin 90^\circ = 1$, $\cos 90^\circ = 0$, $\tan 90^\circ = \text{undefined}$

This comprehensive theory covers all aspects of trigonometry with detailed explanations and examples, providing complete preparation for the entrance examination.