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Ch 03 - Trigonometric Functions11th Mathematics

Geometric Convention for Positive and Negative Angles

What geometric convention strictly determines whether an angle is mathematically positive or negative?

DEFINITION

Core Definition

Positive Angle: Formed by an anticlockwise rotation of the ray from its initial side to the terminal side.
Negative Angle: Formed by a clockwise rotation of the ray from its initial side to the terminal side.

Boundary Conditions / Domain

The initial side is strictly fixed as the reference axis (typically the positive x-axis in a Cartesian system).

Prism Memory Hook

Anti-Clock is Pro-Positive (AC = +)

Arc Length and Radian Measure

APPLICATION Formula

l = r \theta

State the formula relating the length of an arc, the radius of the circle, and the angle subtended at the center.

Derivation

Verify the angle is given in radians. If given in degrees, strictly convert to radians first.
Multiply the radius by the radian angle to compute the arc length directly.

Component Anatomy

$l$ : Length of the intercepted arc.
$r$ : Radius of the circle.
$\theta$ : Subtended central angle.

Boundary Conditions / Domain

$\theta$ MUST be strictly measured in radians, not degrees.
$r > 0$ .

Prism Memory Hook

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Degree and Radian Conversion Formula

APPLICATION Formula

\pi \text{ radian} = 180^{\circ}

State the fundamental equivalence equation connecting degree measure and radian measure.

Derivation

Degrees to Radians: Multiply the degree value by $\frac{\pi}{180}$ .
Radians to Degrees: Multiply the radian value by $\frac{180}{\pi}$ .

Component Anatomy

$\pi$ radian: Exact mathematical equivalent to $180^{\circ}$ .

Boundary Conditions / Domain

Angles can be any real number ( $-\infty < \theta < \infty$ ).

Prism Memory Hook

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Also in this chapter

48 more cards to study in Prism

4Approximate Values of 1 Radian and 1 Degree
5Zeroes of Sine and Cosine Functions
6Domain and Range of Sine and Cosine
7Domain and Range of Tangent Function
8Domain and Range of Cotangent Function
9Domain and Range of Secant and Cosecant
10Asymptotic Behavior of Tangent and Cotangent Graphs
11Signs of Trigonometric Functions Across Quadrants
12Even and Odd Nature of Sine and Cosine
13Complementary Angles (First Quadrant Aliases)
14Supplementary Angles (Second Quadrant Aliases)
15Cosine of Sum of Two Angles
16Cosine of Difference of Two Angles
17Sine of Sum of Two Angles
18Sine of Difference of Two Angles
19Tangent of Sum of Two Angles
20Tangent of Difference of Two Angles
21Cotangent of Sum of Two Angles
22Cotangent of Difference of Two Angles
23Double Angle Identity for Cosine
24Double Angle Identity for Sine
25Double Angle Identity for Tangent
26Triple Angle Identity for Sine
27Triple Angle Identity for Cosine
28Triple Angle Identity for Tangent
29Sum to Product Transformations (Cosine)
30Sum to Product Transformations (Sine)
31Product to Sum Transformations (Cosine & Sine pairs)
32Product to Sum Transformations (Mixed pairs)
33Difference of Squares Trigonometric Identities
34Algebraic Reduction of High-Power Trig Sums
35Principal Solutions of Trigonometric Equations
36General Solutions of Trigonometric Equations
37General Solution for Sine Function
38General Solution for Cosine Function
39General Solution for Tangent Function
40SOP: Transforming a cos(θ) + b sin(θ)
41Extrema of a cos(θ) + b sin(θ)
42Law of Sines (Sine Rule)
43Law of Cosines (Cosine Rule)
44Napier's Analogy (Law of Tangents)
45Half-Angle Formula for Sine (Properties of Triangles)
46Half-Angle Formula for Cosine (Properties of Triangles)
47Half-Angle Formula for Tangent (Properties of Triangles)
48Exact Values of 15 and 75 Degrees
49Exact Values of 18 and 36 Degrees (JEE Edge-Case)
50Product Series of Cosines: Folding Step
51Bounding Trigonometric Expressions using AM-GM Inequality

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11th Mathematics

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