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Detailed Notes - Six Ratios - Standard Values - Complementary Angles

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.

1

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
OABthetaP(x,y)Mryx

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

2

The Six Trigonometric Ratios

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

Reference Triangle
thetaACBPerpendicularBaseHypotenuse

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⁡θ=PH\sin\theta=\frac{P}{H}
cos⁡θ=BH\cos\theta=\frac{B}{H}
tan⁡θ=PB\tan\theta=\frac{P}{B}
cosec⁡θ=HP\cosec\theta=\frac{H}{P}
sec⁡θ=HB\sec\theta=\frac{H}{B}
cot⁡θ=BP\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=35,cos⁡A=45,tan⁡A=34\sin A=\frac35,\quad \cos A=\frac45,\quad \tan A=\frac34

cosec⁡A=53,sec⁡A=54,cot⁡A=43\cosec A=\frac53,\quad \sec A=\frac54,\quad \cot A=\frac43

3

Relations Between Ratios

Reciprocal and quotient identities that help recover all missing ratios.

Reciprocal Relations
cosec⁡θ=1sin⁡θ,sec⁡θ=1cos⁡θ,cot⁡θ=1tan⁡θ\cosec\theta=\frac{1}{\sin\theta},\quad \sec\theta=\frac{1}{\cos\theta},\quad \cot\theta=\frac{1}{\tan\theta}
Quotient Relations
tan⁡θ=sin⁡θcos⁡θ,cot⁡θ=cos⁡θsin⁡θ\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\cosec A=2, then

sin⁡A=12\sin A=\frac12

Take perpendicular =1=1 and hypotenuse =2=2. Then base =3=\sqrt3.

cos⁡A=32,tan⁡A=13\cos A=\frac{\sqrt3}{2},\qquad \tan A=\frac{1}{\sqrt3}

4

Standard Values

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

Standard Value Table
Ratio0 degree30 degree45 degree60 degree90 degree
sin theta01/21/sqrt2sqrt3/21
cos theta1sqrt3/21/sqrt21/20
tan theta01/sqrt31sqrt3Not defined
cosec thetaNot defined2sqrt22/sqrt31
sec theta12/sqrt3sqrt22Not defined
cot thetaNot definedsqrt311/sqrt30
Evaluation Example

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

Show solution

=32⋅32+12⋅12=34+14=1=\frac{\sqrt3}{2}\cdot\frac{\sqrt3}{2}+\frac12\cdot\frac12=\frac34+\frac14=1

5

Complementary Angles

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

Complementary Rules
sin⁡(90∘−θ)=cos⁡θ,cos⁡(90∘−θ)=sin⁡θ\sin(90^\circ-\theta)=\cos\theta,\qquad \cos(90^\circ-\theta)=\sin\theta
tan⁡(90∘−θ)=cot⁡θ,cot⁡(90∘−θ)=tan⁡θ\tan(90^\circ-\theta)=\cot\theta,\qquad \cot(90^\circ-\theta)=\tan\theta
sec⁡(90∘−θ)=cosec⁡θ,cosec⁡(90∘−θ)=sec⁡θ\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,

tan⁡54∘=cot⁡36∘andsin⁡20∘=cos⁡70∘\tan54^\circ=\cot36^\circ\quad\text{and}\quad \sin20^\circ=\cos70^\circ

So the whole expression becomes

1+1−2=01+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

Trigonometric Ratios: revision sheet

Close the notes and explain each idea in your own words. Then check the formulas and attempt the examples before opening their solutions.

Recall checklist
  • Setting Up the Right Triangle
  • The Six Trigonometric Ratios
  • Standard Values — The Table You Must Know
  • Reciprocal, Quotient, and Complementary Angle Relations
  • Trigonometric Identities
  • Common Mistakes and Board Answer Template

For each topic: state the rule, explain when it applies, give an example and identify one mistake to avoid. Return to any topic you cannot explain without looking.

Formula recap (6)

The Six Trigonometric Ratios

sin⁡θ=opphyp,cos⁡θ=adjhyp,tan⁡θ=oppadj\sin\theta = \frac{\text{opp}}{\text{hyp}},\quad \cos\theta = \frac{\text{adj}}{\text{hyp}},\quad \tan\theta = \frac{\text{opp}}{\text{adj}}
Primary ratios. Remember SOH-CAH-TOA.

The Six Trigonometric Ratios

csc⁡θ=1sin⁡θ,sec⁡θ=1cos⁡θ,cot⁡θ=1tan⁡θ\csc\theta = \frac{1}{\sin\theta},\quad \sec\theta = \frac{1}{\cos\theta},\quad \cot\theta = \frac{1}{\tan\theta}
Reciprocal ratios — cosecant, secant, cotangent.

Reciprocal, Quotient, and Complementary Angle Relations

tan⁡θ=sin⁡θcos⁡θ,cot⁡θ=cos⁡θsin⁡θ\tan\theta = \frac{\sin\theta}{\cos\theta},\quad \cot\theta = \frac{\cos\theta}{\sin\theta}
Quotient relations.

Reciprocal, Quotient, and Complementary Angle Relations

sin⁡θ=cos⁡(90∘−θ),tan⁡θ=cot⁡(90∘−θ),sec⁡θ=csc⁡(90∘−θ)\sin\theta = \cos(90^\circ - \theta),\quad \tan\theta = \cot(90^\circ - \theta),\quad \sec\theta = \csc(90^\circ - \theta)
Complementary angle identities — each ratio equals its co-ratio of the complementary angle.

Trigonometric Identities

sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1
Fundamental Pythagorean identity.

Trigonometric Identities

1+tan⁡2θ=sec⁡2θ,1+cot⁡2θ=csc⁡2θ1 + \tan^2\theta = \sec^2\theta,\quad 1 + \cot^2\theta = \csc^2\theta
Derived identities — obtained by dividing the fundamental identity by cos²θ or sin²θ.

Explain your method

Write the given information, choose a method, show the steps, then check units and whether the answer makes sense.

If sin A = 3/5, find cos A and tan A.
Check worked solution 1
Take perpendicular = 3 and hypotenuse = 5. Then base = 4 by Pythagoras, so cos A = 4/5 and tan A = 3/4.
Evaluate tan 54 degrees divided by cot 36 degrees.
Check worked solution 2
Using complementary angles, tan(54 degrees) = cot(36 degrees), so the value is 1.

Turn mistakes into revision

Record the question, the step that went wrong and the corrected reason. Try it again tomorrow without the solution, then revisit it after a week. A correct guess still needs an explanation.

Practise this chapter