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
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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
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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.
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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
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