10th Class Trigonometry Formulas

1. Basic Trigonometric Ratios

In a Right-Angled Triangle:

P = Perpendicular (Opposite Side)

B = Base (Adjacent Side)

H = Hypotenuse

Pythagoras Theorem:

\[P^2 + B^2 = H^2\]

Six Trigonometric Ratios

\(\sin\theta = \frac{\text{Perpendicular}}{\text{Hypotenuse}} = \frac{P}{H}\)
\(\cos\theta = \frac{\text{Base}}{\text{Hypotenuse}} = \frac{B}{H}\)
\(\tan\theta = \frac{\text{Perpendicular}}{\text{Base}} = \frac{P}{B}\)
\(\csc\theta = \frac{\text{Hypotenuse}}{\text{Perpendicular}} = \frac{H}{P}\)
\(\sec\theta = \frac{\text{Hypotenuse}}{\text{Base}} = \frac{H}{B}\)
\(\cot\theta = \frac{\text{Base}}{\text{Perpendicular}} = \frac{B}{P}\)

2. Reciprocal Identities

\(\sin\theta = \frac{1}{\csc\theta}\) or \(\sin\theta \cdot \csc\theta = 1\)
\(\cos\theta = \frac{1}{\sec\theta}\) or \(\cos\theta \cdot \sec\theta = 1\)
\(\tan\theta = \frac{1}{\cot\theta}\) or \(\tan\theta \cdot \cot\theta = 1\)

3. Quotient Identities

\(\tan\theta = \frac{\sin\theta}{\cos\theta}\)
\(\cot\theta = \frac{\cos\theta}{\sin\theta}\)

4. Pythagorean Identities (Most Important)

These are the THREE GOLDEN FORMULAS of Trigonometry:

First Pythagorean Identity

\[\sin^2\theta + \cos^2\theta = 1\]

Derived forms:

\(\sin^2\theta = 1 - \cos^2\theta\)
\(\cos^2\theta = 1 - \sin^2\theta\)

Second Pythagorean Identity

\[1 + \tan^2\theta = \sec^2\theta\]

Derived forms:

\(\tan^2\theta = \sec^2\theta - 1\)
\(\sec^2\theta - \tan^2\theta = 1\)

Third Pythagorean Identity

\[1 + \cot^2\theta = \csc^2\theta\]

Derived forms:

\(\cot^2\theta = \csc^2\theta - 1\)
\(\csc^2\theta - \cot^2\theta = 1\)

5. Complementary Angles (90° - θ)

Key Concept: For complementary angles, sine and cosine interchange, tangent and cotangent interchange, secant and cosecant interchange.

\(\sin(90° - \theta) = \cos\theta\)
\(\cos(90° - \theta) = \sin\theta\)
\(\tan(90° - \theta) = \cot\theta\)
\(\cot(90° - \theta) = \tan\theta\)
\(\sec(90° - \theta) = \csc\theta\)
\(\csc(90° - \theta) = \sec\theta\)

6. Standard Angle Values (Trigonometric Table)

Angle (θ)	0°	30°	45°	60°	90°
sin θ	0	\(\frac{1}{2}\)	\(\frac{1}{\sqrt{2}}\) or \(\frac{\sqrt{2}}{2}\)	\(\frac{\sqrt{3}}{2}\)	1
cos θ	1	\(\frac{\sqrt{3}}{2}\)	\(\frac{1}{\sqrt{2}}\) or \(\frac{\sqrt{2}}{2}\)	\(\frac{1}{2}\)	0
tan θ	0	\(\frac{1}{\sqrt{3}}\) or \(\frac{\sqrt{3}}{3}\)	1	\(\sqrt{3}\)	∞ (Not Defined)
cot θ	∞ (Not Defined)	\(\sqrt{3}\)	1	\(\frac{1}{\sqrt{3}}\) or \(\frac{\sqrt{3}}{3}\)	0
sec θ	1	\(\frac{2}{\sqrt{3}}\) or \(\frac{2\sqrt{3}}{3}\)	\(\sqrt{2}\)	2	∞ (Not Defined)
csc θ	∞ (Not Defined)	2	\(\sqrt{2}\)	\(\frac{2}{\sqrt{3}}\) or \(\frac{2\sqrt{3}}{3}\)	1

Memory Trick for Sine Values:

sin 0° = \(\sqrt{0}/2 = 0\)

sin 30° = \(\sqrt{1}/2 = 1/2\)

sin 45° = \(\sqrt{2}/2\)

sin 60° = \(\sqrt{3}/2\)

sin 90° = \(\sqrt{4}/2 = 1\)

For cosine, reverse the order!

7. Sign Functions (Negative Angles)

\(\sin(-\theta) = -\sin\theta\) (Odd function)
\(\cos(-\theta) = \cos\theta\) (Even function)
\(\tan(-\theta) = -\tan\theta\) (Odd function)
\(\cot(-\theta) = -\cot\theta\) (Odd function)
\(\sec(-\theta) = \sec\theta\) (Even function)
\(\csc(-\theta) = -\csc\theta\) (Odd function)

8. Periodic Identities

First Quadrant (0° to 90°)

\(\sin(90° - \theta) = \cos\theta\)
\(\cos(90° - \theta) = \sin\theta\)

Second Quadrant (90° to 180°)

\(\sin(90° + \theta) = \cos\theta\)
\(\cos(90° + \theta) = -\sin\theta\)
\(\sin(180° - \theta) = \sin\theta\)
\(\cos(180° - \theta) = -\cos\theta\)

Third Quadrant (180° to 270°)

\(\sin(180° + \theta) = -\sin\theta\)
\(\cos(180° + \theta) = -\cos\theta\)

Fourth Quadrant (270° to 360°)

\(\sin(360° - \theta) = -\sin\theta\)
\(\cos(360° - \theta) = \cos\theta\)

9. Maximum and Minimum Values

Function	Maximum Value	Minimum Value
sin θ	1 (at 90°)	-1 (at 270°)
cos θ	1 (at 0°, 360°)	-1 (at 180°)
tan θ	∞	-∞
cot θ	∞	-∞
sec θ	∞	1
csc θ	∞	1

10. Special Formulas

Sum and Product Formulas (Class 10 Level)

\(\sin A \cos B + \cos A \sin B = \sin(A + B)\)
\(\cos A \cos B - \sin A \sin B = \cos(A + B)\)
\(\sin 2A = 2\sin A \cos A\)
\(\cos 2A = \cos^2 A - \sin^2 A = 2\cos^2 A - 1 = 1 - 2\sin^2 A\)

11. Important Properties

Range: -1 ≤ sin θ ≤ 1 and -1 ≤ cos θ ≤ 1
Period: sin and cos have period 360°, tan and cot have period 180°
At 0°: sin 0° = 0, cos 0° = 1, tan 0° = 0
At 90°: sin 90° = 1, cos 90° = 0, tan 90° = undefined
Quadrant rule: All Students Take Calculus (All, Sin, Tan, Cos are positive in I, II, III, IV quadrants)

12. Quick Reference Summary

Category	Key Formulas
Basic Ratios	sin, cos, tan, cot, sec, csc
Reciprocal	sin·csc = 1, cos·sec = 1, tan·cot = 1
Quotient	tan = sin/cos, cot = cos/sin
Pythagorean	sin²+cos²=1, 1+tan²=sec², 1+cot²=csc²
Complementary	sin(90°-θ)=cosθ, cos(90°-θ)=sinθ