Balaji Krafts & Tutorials

Class 11 Mathematics • Tutor: Pradeep Sir

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Concept Study Syllabus

Trigonometric Functions & Angle Measures

Welcome to the interactive digital smart-board. This infographic syllabus covers circular angle systems, allied angle matrices, compound formula derivations, and multiples. Click the topics to explore Pradeep Sir's interactive notebooks or play with the unit circle simulator below!

Pradeep Sir's Tutor Corner

"Visualizing trigonometry makes formulas intuitive. Try rotating the slider in the Unit Circle simulator to see how coordinates translate instantly into cosine and sine!"

Chapter Infographics Dashboard

01 Measurement

Measurement of Angles

Understand Sexagesimal measures ($D^\circ M' S''$), Circular measure radians, and the fundamental constant angle proof of $1^c$.

Explore Notebook

02 Circular

Circular Functions

See circular coordinates $P(x,y) = (\cos\theta, \sin\theta)$ inside a unit radius circle. Covers ASTC quadrant systems, domains & ranges, and periodic states.

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03 Formulas

Compound & Allied Angles

Derive trigonometric functions of sum & difference of numbers ($\cos(A\pm B)$) and matrices of allied functions ($\pi/2 \pm x$, $\pi \pm x$).

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04 Multiples

Multiples & Submultiples

Explore double and triple angle identities ($2x$, $3x$, $x/2$), derivations, and special values ($\sin 18^\circ$, $\cos 36^\circ$).

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Real-time Unit Circle Simulator

Drag slider or input an angle to test initial & terminal side coordinates.

Simulator

Terminal Position P(x, y):

P(0.707, 0.707)

Angle Controls

-360° (Clockwise) 720° (Multi-rotation)

Radian Measure

Active Quadrant

Quadrant I

$\sin\theta$ (y coordinate): 0.7071

$\cos\theta$ (x coordinate): 0.7071

$\tan\theta$ ($y/x$ ratio): 1.0000

Smart Converter & Formula Explorer

Explore conversions and compound addition/subtraction calculations dynamically.

Degree-Radian Tool

Degree °

Radian Measure

Compound $A \pm B$ Calculator

Function

Operator

Angle A (°)

Angle B (°)

Theorem Proof Path

Calculated Output:

Topic 3: Sum, Difference & Allied Angles

Geometric Transformations and Derivations

Trigonometric additions represent spatial transformations along the circular plane. Geometrical proof derives the product relations between point parameters.

Compound Addition Formulas

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

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

$\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$

$\cot(A \pm B) = \frac{\cot A \cot B \mp 1}{\cot B \pm \cot A}$

Allied Transformation Table

Ratio	$-x$	$\pi/2 - x$	$\pi/2 + x$	$\pi - x$	$\pi + x$
$\sin$	$-\sin x$	$\cos x$	$\cos x$	$\sin x$	$-\sin x$
$\cos$	$\cos x$	$\sin x$	$-\sin x$	$-\cos x$	$-\cos x$
$\tan$	$-\tan x$	$\cot x$	$-\cot x$	$-\tan x$	$\tan x$

Worked Examples Notebook

Example 1: Compounding Values ($\sin 75^\circ$)

Question: Evaluate the exact numerical value of $\sin 75^\circ$.
Step-by-step Solution:
Decompose the angle into standard unit boundaries: $75^\circ = 45^\circ + 30^\circ$.
Apply the compound addition formulation: $\sin(A+B) = \sin A\cos B + \cos A\sin B$. $$\sin(45^\circ + 30^\circ) = \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ$$ Substitute the standard exact values: $$\sin 75^\circ = \left(\frac{1}{\sqrt{2}}\right)\left(\frac{\sqrt{3}}{2}\right) + \left(\frac{1}{\sqrt{2}}\right)\left(\frac{1}{2}\right) = \frac{\sqrt{3}+1}{2\sqrt{2}}$$

Example 2: Proof of Fractional Allied Identities

Question: Prove that $\frac{\cos 9^\circ + \sin 9^\circ}{\cos 9^\circ - \sin 9^\circ} = \tan 54^\circ$.
Step-by-step Solution:
Divide both the numerator and the denominator by $\cos 9^\circ$: $$\text{LHS } = \frac{\frac{\cos 9^\circ}{\cos 9^\circ} + \frac{\sin 9^\circ}{\cos 9^\circ}}{\frac{\cos 9^\circ}{\cos 9^\circ} - \frac{\sin 9^\circ}{\cos 9^\circ}} = \frac{1 + \tan 9^\circ}{1 - \tan 9^\circ}$$ Substitute $1 = \tan 45^\circ$: $$\text{LHS } = \frac{\tan 45^\circ + \tan 9^\circ}{1 - \tan 45^\circ\tan 9^\circ} = \tan(45^\circ + 9^\circ) = \tan 54^\circ \quad (\text{Proved})$$

Example 3: Sum-to-Product Transformation

Question: Express $\cos 7x + \cos 5x$ as a product of trigonometric ratios.
Step-by-step Solution:
Using the sum-to-product formula: $\cos C + \cos D = 2\cos\left(\frac{C+D}{2}\right)\cos\left(\frac{C-D}{2}\right)$.
Set $C = 7x$ and $D = 5x$: $$\cos 7x + \cos 5x = 2\cos\left(\frac{7x+5x}{2}\right)\cos\left(\frac{7x-5x}{2}\right)$$ $$\cos 7x + \cos 5x = 2\cos(6x)\cos(x)$$

Trigonometric Functions & Angle Measures

Chapter Infographics Dashboard

Measurement of Angles

Circular Functions

Compound & Allied Angles

Multiples & Submultiples

Real-time Unit Circle Simulator

Smart Converter & Formula Explorer

Smart Learning Progress Evaluation

Topic 1: Measurement of Angles

Angles Systems Overview

Constant Angle Theorem

Central Angle Equation: $\theta = \frac{l}{r}$

Worked Examples Notebook

Topic 2: Circular Trigonometric Functions

Fundamental Identities Proof

Signs of T-Functions in Various Quadrants

Worked Examples Notebook

Topic 3: Sum, Difference & Allied Angles

Compound Addition Formulas

Allied Transformation Table

Worked Examples Notebook

Topic 4: Multiples, Submultiples & Special Values

Double & Triple Angle Equations

Half Angle / Submultiple Equations

Worked Examples Notebook