# 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:** $$\theta = 360^\circ$$

***

### **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:**

* **S**ine = **O**pposite/**H**ypotenuse
* **C**osine = **A**djacent/**H**ypotenuse
* **T**angent = **O**pposite/**A**djacent

#### **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                      | $$\infty$$             |

***

### **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. $$\tan(\theta + \pi) = \tan\theta$$

***

### **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"

* **A**ll: All positive in Quadrant I
* **S**tudents: Sine positive in Quadrant II
* **T**ake: Tangent positive in Quadrant III
* **C**alculus: 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:** $$\sin 210^\circ = -\sin 30^\circ = -\frac{1}{2}$$

***

### **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**

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

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

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

***

### **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**

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

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

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

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

***

### **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: $$\theta = 2n\pi \pm \frac{2\pi}{3}$$

***

### **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]$$

***

### **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}$$

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

***

### **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 $$B$$:

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

***

### **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

***

### **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

***

### **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**

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

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

***

### **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: $$x = 0$$, $$\frac{\pi}{3}$$, $$\frac{5\pi}{3}$$

***

### **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.