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Trigonometric Ratios: sin, cos, tan Without the Panic

This chapter keeps our shared notes UI while covering the six trigonometric ratios, standard values, complementary angles, and the most common Class 10 evaluation patterns with diagrams and solved examples.

Introduction and Angle Setup

What trigonometry measures and how a right triangle stores the ratio information.

Trigonometry is the branch of mathematics that connects the sides and angles of triangles. In Class 10, the focus is on right triangles and the six ratios made from base, perpendicular, and hypotenuse.

Think of standing at a distance from a tall tower and looking up. If you know the angle and one side, trigonometry helps you uncover the missing length.

Angle and Right Triangle Setup

Point P on the terminal ray creates the right triangle used for the six trigonometric ratios.

The Six Trigonometric Ratios

Definitions of sin, cos, tan, cosec, sec, and cot.

Reference Triangle

For angle theta at A, the side opposite the angle is the perpendicular, the adjacent side is the base, and the longest side is the hypotenuse.

Core Definitions

\sin\theta=\frac{P}{H}

\cos\theta=\frac{B}{H}

\tan\theta=\frac{P}{B}

\cosec\theta=\frac{H}{P}

\sec\theta=\frac{H}{B}

\cot\theta=\frac{B}{P}

Solved Example

For angle A in a right triangle, base = 4, perpendicular = 3, and hypotenuse = 5. Find all six trigonometric ratios.

Show solution

\\sin A=\\frac35,\\quad \\cos A=\\frac45,\\quad \\tan A=\\frac34

\\cosec A=\\frac53,\\quad \\sec A=\\frac54,\\quad \\cot A=\\frac43

Relations Between Ratios

Reciprocal and quotient identities that help recover all missing ratios.

Reciprocal Relations

\\cosec\\theta=\\frac{1}{\\sin\\theta},\\quad \\sec\\theta=\\frac{1}{\\cos\\theta},\\quad \\cot\\theta=\\frac{1}{\\tan\\theta}

Quotient Relations

\\tan\\theta=\\frac{\\sin\\theta}{\\cos\\theta},\\qquad \\cot\\theta=\\frac{\\cos\\theta}{\\sin\\theta}

Identity Example

If cosec A = 2, find sin A, cos A, and tan A.

Show solution

If $\\cosec A=2$ , then

\\sin A=\\frac12

Take perpendicular $=1$ and hypotenuse $=2$ . Then base $=\\sqrt3$ .

\\cos A=\\frac{\\sqrt3}{2},\\qquad \\tan A=\\frac{1}{\\sqrt3}

Standard Values

The must-know table for 0, 30, 45, 60, and 90 degrees.

Standard Value Table

Ratio	0 degree	30 degree	45 degree	60 degree	90 degree
sin theta	0	1/2	1/sqrt2	sqrt3/2	1
cos theta	1	sqrt3/2	1/sqrt2	1/2	0
tan theta	0	1/sqrt3	1	sqrt3	Not defined
cosec theta	Not defined	2	sqrt2	2/sqrt3	1
sec theta	1	2/sqrt3	sqrt2	2	Not defined
cot theta	Not defined	sqrt3	1	1/sqrt3	0

Evaluation Example

Evaluate sin 60 degree cos 30 degree + cos 60 degree sin 30 degree.

Show solution

=\\frac{\\sqrt3}{2}\\cdot\\frac{\\sqrt3}{2}+\\frac12\\cdot\\frac12=\\frac34+\\frac14=1

Complementary Angles

The 90 degree minus theta rules used in simplification and board problems.

Complementary Rules

\\sin(90^\\circ-\\theta)=\\cos\\theta,\\qquad \\cos(90^\\circ-\\theta)=\\sin\\theta

\\tan(90^\\circ-\\theta)=\\cot\\theta,\\qquad \\cot(90^\\circ-\\theta)=\\tan\\theta

\\sec(90^\\circ-\\theta)=\\cosec\\theta,\\qquad \\cosec(90^\\circ-\\theta)=\\sec\\theta

Complementary Example

Evaluate tan 54 degree divided by cot 36 degree plus sin 20 degree divided by cos 70 degree minus 2.

Show solution

Using complementary angles,

\\tan54^\\circ=\\cot36^\\circ\\quad\\text{and}\\quad \\sin20^\\circ=\\cos70^\\circ

So the whole expression becomes

1+1-2=0

Trigonometric Ratios Summary

sin theta

Perpendicular / Hypotenuse

cos theta

Base / Hypotenuse

tan theta

Perpendicular / Base

cosec theta

1 / sin theta

sec theta

1 / cos theta

cot theta

1 / tan theta

Complementary rule

sin(90° - theta) = cos theta

Exam habit

Find base, perpendicular, and hypotenuse first.

Open Practice

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