World-class educational books. Written, reviewed, and designed by AI.
Pageforce Production is the system publishing companies run in-house to produce educational books in days instead of months — in their own house style, with their own editors in command.
Ratios that bind an angle to the sides of a right triangle.
✓
Theory
In a right-angled triangle, the six trigonometric ratios fix the relationship between an acute angle and the lengths of its three sides — the side opposite , the side adjacent to it, and the hypotenuse.
✓
Formula
4.1 The three primary ratios
✓
Figure
Fig. 4.1 — sides named relative to .
Each ratio is a pure number: it depends on the angle alone, never on the size of the triangle. The reciprocals give cosecant, secant and cotangent.
✓
Table
θ
0°
30°
45°
60°
90°
sin θ
0
½
1
cos θ
1
½
0
tan θ
0
1
∞
✓
Example
Example 1.
If , find and .
Solution. With opp , hyp , the adjacent side is . Hence and .
✓
MCQ
Ex 4.1 The value of is
A
B
C
D
✓
Topper Tip✓
PYQPrevious Years' QuestionsNDA · 2018–2023
NDA 2019 If , then equals
Ans. Squaring, , so it is .
NDA 2021 The value of is
Ans. .
✓
42Trigonometry
Live — one chapter: Outline → Writing → Review → Press-ready page
No designer touched this page
How it works
Approve the plan. Pageforce does the rest.
1✓OutlineYour editorYour editor approves the plan
2✓WritingAIAI writes every section
3✓ReviewAIEvery page is checked
4✓DesignAIPress-ready pages come out
One chapter, end to end — your editor approves, the machine does the rest
Watch it happen — in the real product.
Pageforce Studio
OutlineWritingReviewDesign
Chapter outline
Chapter 4 — Trigonometric Ratios
7 sections · approved structure
1Introduction to ratiosconcept · 3 paras
2The six ratios defined2 examples · 4 figures
3Values of standard angles1 table · 3 examples
4Worked examples2 examples
5Practice set 4.16 MCQs
6Previous-year questions4 PYQs
7Summary & formula box1 formula box
OutlineWritingReviewDesign
Section 2 of 7 · The six ratios defined0 words
In a right-angled triangle, the six trigonometric ratios describe the fixed relationship between an angle and the lengths of its three sides.
The side opposite the angle, the side adjacent to it, and the hypotenuse combine, two at a time, to define sine, cosine and tangent.
Because each ratio depends only on the angle — never on the size of the triangle — the same value holds for every similar triangle.
This is what lets us recover distances we cannot measure directly, from the height of a tower to the width of a river.
OutlineWritingReviewDesign
Review
11 automated checks
Syllabus coverage
Mathematical accuracy
Voice consistency
MCQ answer correctness
Difficulty distribution
Formula coverage
Structural completeness
…and 4 more
OutlineWritingReviewDesign
## Chapter 4 — Trigonometric Ratios
In a right-angled triangle, the six **trigonometric ratios** describe the fixed relationship between an angle and its sides.
$$ \sin\theta / \cos\theta = \tan\theta $$
**MCQ 1.** The value of sin 30° is (a) 1/2 (b) √3/2 (c) 1 (d) 0
NDA · MathematicsTrigonometry
Chapter 4 — Trigonometric Ratios
In a right-angled triangle, the six trigonometric ratios describe the fixed relationship between an angle and the lengths of its sides. Because each ratio depends only on the angle, the same values hold for every similar triangle.
This is what lets us recover distances we cannot measure directly — the height of a tower, the width of a river — from a single measured angle.
42Trigonometry
This is not a rendering. This is the product.
The output
From raw content to fully designed page you could print tomorrow
Every AI can write. Only Pageforce designs the book. Formulas, figures, tables, question grids, folios — typeset automatically, to press standard. No other system in the world does this.
chapter-04.md
## Chapter 4 — Trigonometric Ratios
In a right-angled triangle, the six **trigonometric ratios**
describe the fixed relationship between an angle and its sides.
$$\sin\theta=\frac{opp}{hyp},\ \cos\theta=\frac{adj}{hyp}$$**MCQ 1.** The value of sin 30° is(a) 1/2 (b) √3/2
(c) 1 (d) 0
> Topper Tip: memorise Pandit Badri Prasad.
4 · Trigonometric RatiosMathematics — Class X
4
Chapter Four
Trigonometric Ratios of an Acute Angle
Ratios that bind an angle to the sides of a right triangle — and the standard values every problem returns to.
In a right-angled triangle, the six trigonometric ratios fix the relationship between an acute angle and the lengths of its three sides — the side opposite , the side adjacent to it, and the hypotenuse. Every identity, height-and-distance problem, and graph that follows in this unit is a consequence of these three definitions, so they are worth committing to memory before proceeding.
4.1 The three primary ratios
For the reference angle shown in Fig. 4.1, the primary ratios are defined as follows. Each is a pure number, independent of the size of the triangle.
Fig. 4.1 — Sides named relative to ; here .
The reciprocals give three further ratios — cosecant, secant and cotangent — defined by , and . Together the six ratios describe the triangle completely.
4.2 Standard values
The ratios at the standard angles recur constantly and should be recalled instantly:
θ
0°
30°
45°
60°
90°
sin θ
0
½
1
cos θ
1
½
0
tan θ
0
1
∞
Example 1.
If , find and .
Solution. With opp , hyp , the adjacent side is . Hence and .
Ex 4.1 The value of is
A
B
C
D
4.3 Fundamental identities
Applying the Pythagorean theorem to the definitions of the ratios yields the three identities on which most simplifications depend. Dividing throughout by , and in turn gives:
These hold for every angle for which each ratio is defined, and together with the quotient relations and they reduce any trigonometric expression to sines and cosines alone.
Example 2.
Prove that .
Solution. Since , the left side is . ∎
Previous Years' QuestionsNDA · 2018–2023
NDA 2019 If , then equals
Ans. Squaring, , so .
NDA 2021 The value of is
Ans. .
NDA 2022 A tower subtends at a point away. Its height is
Ans. .
NDA 2023 If , the value of is
Ans. opp , so .
42Trigonometry
Left: raw manuscript · Right: press-ready page — same content
No designer touched this page
No designer touched this page.
The publisher engine
First it learns your books. Then everything it makes is yours.
Show Pageforce your best titles. It learns how you write, how you teach, and how your pages look — and produces new books that match them. Nothing generic. Nothing off-brand.
Same series. Same standard. The right-hand three did not exist until Pageforce made them.
Your voiceevery sentence sounds like your house wrote it
Your designyour layouts, your covers-to-folios visual identity
Your pedagogyyour way of teaching, your difficulty curve
Your standardsyour review bar, enforced on every page
Your competitors can buy the same AI models. They cannot buy your engine.
Measured, not promised
Two months of production. Done in two days.
01Turnaround
60 days
12
2 months →2 daysOne chapter, outline to press-ready pages.
02Throughput
0 pages00:00
17 pages in 16 minutesA complete designed chapter, in a single run.
Built for
For publishers of competitive-exam and curriculum books.
Pageforce Production is an enterprise system for educational publishing houses — the companies producing NDA, SSC, banking, JEE, and school curriculum titles. It is not a consumer writing tool. It is the production floor your house runs in-house, operated by your own editorial team.
If that is you, we should talk.
The suite
Production is live today. Three more machines are coming.
pageforceEdit2026
Hand it a manuscript, get back a professionally edited one. Your voice intact.
The results was really very good andwere decisive, and confirmed the hypothesis.
pageforceDesign2026
A finished draft becomes a fully typeset book. Minutes, not months.
It was the season of light, it was the season of darkness.
It was the season of light, it was the season of darkness.
pageforceTranslate2027
Your entire catalog, in every Indian language. Book for book, not word for word.
Knowledge belongs in every language.
ज्ञान हर भाषा में होना चाहिए।
One company, covering the whole life of a book.
Bring us one chapter. We'll show you the machine.
A working session is simple: you pick a chapter from a book you already publish. We produce it — written, reviewed, and designed — while you watch. Then you compare it to your own edition and decide for yourself.