Unit 5 Trigonometry – अपवर्त्य कोण, गुणनफल रुपान्तरण र त्रिकोणमितीय समीकरण
Multiple Angles, Transformation Formulas, Conditional Identities, Trigonometric Equations & Height-Distance Problems
Introduction to Trigonometry
Unit 5 Trigonometry covers advanced trigonometric concepts including multiple and submultiple angles, transformation of products and sums, conditional trigonometric identities, trigonometric equations, and applications in height and distance problems. These concepts are fundamental for solving complex trigonometric problems in mathematics, physics, and engineering.
1. अपवर्त्य कोणहरू (Multiple Angles)
अपवर्त्य कोण (Multiple Angle)
कुनै दिइएको कोण A को अपवर्त्य कोण भन्नाले 2A, 3A, 4A लगायतको कोणहरूलाई बुझिन्छ। यी कोणहरूको त्रिकोणमितीय अनुपातहरू A को त्रिकोणमितीय अनुपातहरूद्वारा व्यक्त गर्न सकिन्छ।
Multiple angles refer to angles like 2A, 3A, 4A, etc., for a given angle A. Their trigonometric ratios can be expressed in terms of trigonometric ratios of angle A.
द्विक कोणका सूत्रहरू (Double Angle Formulas)
$$sin~2A = 2~sin~A~cos~A$$
$$cos~2A = cos^{2}A – sin^{2}A$$
$$cos~2A = 2~cos^{2}A – 1$$
$$cos~2A = 1 – 2~sin^{2}A$$
$$tan~2A = \frac{2~tan~A}{1 – tan^{2}A}$$
$$1 + cos~2A = 2~cos^{2}A$$
$$1 – cos~2A = 2~sin^{2}A$$
$$\frac{1 – cos~2A}{1 + cos~2A} = tan^{2}A$$
$$sin~2A = \frac{2~tan~A}{1 + tan^{2}A}$$
$$cos~2A = \frac{1 – tan^{2}A}{1 + tan^{2}A}$$
त्रिक कोणका सूत्रहरू (Triple Angle Formulas)
$$sin~3A = 3~sin~A – 4~sin^{3}A$$
$$4~sin^{3}A = 3~sin~A – sin~3A$$
$$cos~3A = 4~cos^{3}A – 3~cos~A$$
$$4~cos^{3}A = 3~cos~A + cos~3A$$
उदाहरण (Example)
यदि $sin~A = \frac{3}{5}$ भए $sin~2A$, $cos~2A$ र $tan~2A$ को मान पत्ता लगाउनुहोस् ।
Solution:
$sin~A = \frac{3}{5}$, तब $cos~A = \frac{4}{5}$ (किनकि $sin^{2}A + cos^{2}A = 1$)
$sin~2A = 2~sin~A~cos~A = 2 \times \frac{3}{5} \times \frac{4}{5} = \frac{24}{25}$
$cos~2A = 1 – 2~sin^{2}A = 1 – 2 \times (\frac{3}{5})^{2} = 1 – \frac{18}{25} = \frac{7}{25}$
$tan~2A = \frac{sin~2A}{cos~2A} = \frac{24/25}{7/25} = \frac{24}{7}$
2. अपवर्तक कोणहरू (Submultiple Angles)
अपवर्तक कोण (Submultiple Angle)
कुनै दिइएको कोण A को अपवर्तक कोण भन्नाले $\frac{A}{2}$, $\frac{A}{3}$, $\frac{A}{4}$ लगायतको कोणहरूलाई बुझिन्छ।
Submultiple angles refer to angles like $\frac{A}{2}$, $\frac{A}{3}$, $\frac{A}{4}$, etc., for a given angle A.
अर्ध कोणका सूत्रहरू (Half Angle Formulas)
$$sin~A = 2~sin\frac{A}{2}~cos\frac{A}{2}$$
$$cos~A = cos^{2}\frac{A}{2} – sin^{2}\frac{A}{2}$$
$$cos~A = 2~cos^{2}\frac{A}{2} – 1$$
$$cos~A = 1 – 2~sin^{2}\frac{A}{2}$$
$$1 + cos~A = 2~cos^{2}\frac{A}{2}$$
$$1 – cos~A = 2~sin^{2}\frac{A}{2}$$
$$tan~A = \frac{2~tan\frac{A}{2}}{1 – tan^{2}\frac{A}{2}}$$
$$\frac{1 – cos~A}{1 + cos~A} = tan^{2}\frac{A}{2}$$
$$sin~A = \frac{2~tan\frac{A}{2}}{1 + tan^{2}\frac{A}{2}}$$
$$cos~A = \frac{1 – tan^{2}\frac{A}{2}}{1 + tan^{2}\frac{A}{2}}$$
एक-तिहाइ कोणका सूत्रहरू (One-Third Angle Formulas)
$$sin~A = 3~sin\frac{A}{3} – 4~sin^{3}\frac{A}{3}$$
$$4~sin^{3}\frac{A}{3} = 3~sin\frac{A}{3} – sin~A$$
$$cos~A = 4~cos^{3}\frac{A}{3} – 3~cos\frac{A}{3}$$
$$4~cos^{3}\frac{A}{3} = 3~cos\frac{A}{3} + cos~A$$
उदाहरण (Example)
यदि $cos~A = \frac{1}{2}$ भए $cos\frac{A}{2}$ को मान पत्ता लगाउनुहोस् ।
Solution:
$cos~A = 2~cos^{2}\frac{A}{2} – 1$
$\frac{1}{2} = 2~cos^{2}\frac{A}{2} – 1$
$2~cos^{2}\frac{A}{2} = \frac{1}{2} + 1 = \frac{3}{2}$
$cos^{2}\frac{A}{2} = \frac{3}{4}$
$cos\frac{A}{2} = \pm \frac{\sqrt{3}}{2}$
(Sign depends on quadrant of $\frac{A}{2}$)
3. गुणन र योगफलको रुपान्तरण (Transformation of Product and Sum)
योगफललाई गुणनफलमा रुपान्तरण (Sum to Product Transformation)
$$sin~C + sin~D = 2~sin(\frac{C+D}{2})~cos(\frac{C-D}{2})$$
$$sin~C – sin~D = 2~cos(\frac{C+D}{2})~sin(\frac{C-D}{2})$$
$$cos~C + cos~D = 2~cos(\frac{C+D}{2})~cos(\frac{C-D}{2})$$
$$cos~C – cos~D = 2~sin(\frac{C+D}{2})~sin(\frac{D-C}{2})$$
नोट: यहाँ $sin(\frac{D-C}{2})$ हो, $sin(\frac{C-D}{2})$ होइन
गुणनफललाई योगफलमा रुपान्तरण (Product to Sum Transformation)
$$2~sin~A~cos~B = sin(A+B) + sin(A-B)$$
$$2~cos~A~sin~B = sin(A+B) – sin(A-B)$$
$$2~cos~A~cos~B = cos(A+B) + cos(A-B)$$
$$2~sin~A~sin~B = cos(A-B) – cos(A+B)$$
उदाहरण (Example)
$sin~75^{\circ} + sin~15^{\circ}$ को मान पत्ता लगाउनुहोस् ।
Solution:
$sin~75^{\circ} + sin~15^{\circ} = 2~sin(\frac{75^{\circ}+15^{\circ}}{2})~cos(\frac{75^{\circ}-15^{\circ}}{2})$
$= 2~sin~45^{\circ}~cos~30^{\circ}$
$= 2 \times \frac{1}{\sqrt{2}} \times \frac{\sqrt{3}}{2}$
$= \frac{\sqrt{3}}{\sqrt{2}} = \frac{\sqrt{6}}{2}$
उदाहरण (Example)
$2~cos~50^{\circ}~cos~10^{\circ}$ लाई योगफलको रूपमा लेख्नुहोस् ।
Solution:
$2~cos~50^{\circ}~cos~10^{\circ} = cos(50^{\circ}+10^{\circ}) + cos(50^{\circ}-10^{\circ})$
$= cos~60^{\circ} + cos~40^{\circ}$
$= \frac{1}{2} + cos~40^{\circ}$
4. अनुबन्धित त्रिकोणमितीय सर्वसमीकाहरू (Conditional Trigonometric Identities)
त्रिभुजका तीनकोणहरूको योग (Sum of Three Angles of a Triangle)
त्रिभुजका तीनकोणहरूको योग $180^{\circ}$ वा $\pi^{c}$ हुन्छ। यदि $A$, $B$, $C$ त्रिभुजका कोणहरू भए $A+B+C = \pi^{c}$ हुन्छ।
Sum of three angles of a triangle is $180^{\circ}$ or $\pi^{c}$. If $A$, $B$, $C$ are angles of a triangle, then $A+B+C = \pi^{c}$.
महत्त्वपूर्ण सर्वसमिकाहरू (Important Identities)
$$A+B = \pi^{c} – C$$
$$sin(A+B) = sin~C$$
$$cos(A+B) = -cos~C$$
$$tan(A+B) = -tan~C$$
$$sin(\frac{A}{2}+\frac{B}{2}) = cos\frac{C}{2}$$
$$cos(\frac{A}{2}+\frac{B}{2}) = sin\frac{C}{2}$$
$$tan(\frac{A}{2}+\frac{B}{2}) = cot\frac{C}{2}$$
$$sin~A + sin~B + sin~C = 4~cos\frac{A}{2}~cos\frac{B}{2}~cos\frac{C}{2}$$
निगमन (Derivation)
$A+B+C = \pi^{c}$ भए $A+B = \pi^{c} – C$
त्यसैले $sin(A+B) = sin(\pi^{c} – C) = sin~C$
$cos(A+B) = cos(\pi^{c} – C) = -cos~C$
$tan(A+B) = tan(\pi^{c} – C) = -tan~C$
यसैगरी $\frac{A}{2}+\frac{B}{2} = \frac{\pi^{c}}{2} – \frac{C}{2}$
$sin(\frac{A}{2}+\frac{B}{2}) = sin(\frac{\pi^{c}}{2} – \frac{C}{2}) = cos\frac{C}{2}$
उदाहरण (Example)
त्रिभुज ABC मा $A+B+C = \pi^{c}$ भए प्रमाणित गर्नुहोस्: $sin~2A + sin~2B + sin~2C = 4~sin~A~sin~B~sin~C$
Proof:
LHS = $sin~2A + sin~2B + sin~2C$
$= 2~sin(A+B)~cos(A-B) + sin~2C$
$= 2~sin(\pi^{c}-C)~cos(A-B) + 2~sin~C~cos~C$
$= 2~sin~C~cos(A-B) + 2~sin~C~cos~C$
$= 2~sin~C[cos(A-B) + cos~C]$
$= 2~sin~C[cos(A-B) + cos(\pi^{c}-(A+B))]$
$= 2~sin~C[cos(A-B) – cos(A+B)]$
$= 2~sin~C \times 2~sin~A~sin~B$
$= 4~sin~A~sin~B~sin~C$ = RHS
5. त्रिकोणमितीय समीकरणहरू (Trigonometric Equations)
चतुर्थांश नियम (Quadrant Rules)
I
All +ve
II
sin +ve
III
tan +ve
IV
cos +ve
ASTC: All Students Take Chemistry (or All Sin Tan Cos)
त्रिकोणमितीय समीकरणहरूको समाधान (Solution of Trigonometric Equations)
| क्र.सं. | समीकरण | अवस्था | चतुर्थांश | मानहरू |
|---|---|---|---|---|
| 1 | $sin~\theta = k$ | $+ve~\& \le 1$ | $1^{st} \& 2^{nd}$ | $\theta, (180^{\circ}-\theta), (360^{\circ}+\theta)$ |
| 2 | $sin~\theta = -k$ | $-ve \ge -1$ | $3^{rd} \& 4^{th}$ | $(180^{\circ}+\theta), (360^{\circ}-\theta)$ |
| 3 | $sin~\theta = k$ | $k$ is non standard | – | $\theta = sin^{-1}k$ |
| 4 | $sin~\theta = k$ | $k > 1$ or $k < -1$ | – | no solution |
| 5 | $cos~\theta = k$ | $+ve~\& \le 1$ | $1^{st} \& 4^{th}$ | $\theta, (360^{\circ}-\theta), (360^{\circ}+\theta)$ |
| 6 | $cos~\theta = -k$ | $-ve \ge -1$ | $2^{nd} \& 3^{rd}$ | $(180^{\circ}-\theta), (180^{\circ}+\theta)$ |
| 7 | $cos~\theta = k$ | $k$ is non standard | – | $\theta = cos^{-1}k$ |
| 8 | $cos~\theta = k$ | $k > 1$ or $k < -1$ | – | no solution |
| 9 | $tan~\theta = k$ | $+ve$ | $1^{st} \& 3^{rd}$ | $\theta, (180^{\circ}+\theta), (360^{\circ}+\theta)$ |
| 10 | $tan~\theta = -k$ | $-ve$ | $2^{nd} \& 4^{th}$ | $(180^{\circ}-\theta), (360^{\circ}-\theta)$ |
| 11 | $tan~\theta = k$ | $k$ is non standard | – | $\theta = tan^{-1}k$ |
उदाहरण 1
$sin~\theta = \frac{1}{2}$ को सामान्य समाधान पत्ता लगाउनुहोस् ।
Solution:
$sin~\theta = \frac{1}{2}$
$sin~\theta = sin~30^{\circ}$
$\theta = 30^{\circ}$ वा $180^{\circ} – 30^{\circ}$
$\theta = 30^{\circ}$ वा $150^{\circ}$
सामान्य समाधान: $\theta = n\pi^{c} + (-1)^{n} \frac{\pi}{6}$
उदाहरण 2
$cos~\theta = -\frac{1}{\sqrt{2}}$ को सामान्य समाधान पत्ता लगाउनुहोस् ।
Solution:
$cos~\theta = -\frac{1}{\sqrt{2}}$
$cos~\theta = -cos~45^{\circ} = cos~135^{\circ}$
$\theta = 135^{\circ}$ वा $225^{\circ}$
सामान्य समाधान: $\theta = 2n\pi^{c} \pm \frac{3\pi}{4}$
6. उचाई र दुरी (Height and Distance)
उचाई र दुरी (Height and Distance)
त्रिकोणमितिको प्रयोग गरेर नाप्न नसकिने वस्तुहरूको उचाई र दुरी पत्ता लगाउन सकिन्छ। यसमा कोणको उन्नतांश र अवनतांशको अवधारणा प्रयोग गरिन्छ।
Trigonometry is used to find heights and distances of inaccessible objects. Concepts of angle of elevation and angle of depression are used.
परिभाषाहरू (Definitions)
उन्नतांश कोण (Angle of Elevation)
कुनै वस्तुलाई हेर्दा तलबाट माथितिर हेर्दा क्षैतिज रेखासँग बनेको कोणलाई उन्नतांश कोण भनिन्छ।
The angle formed with the horizontal line when looking up at an object.
अवनतांश कोण (Angle of Depression)
कुनै वस्तुलाई हेर्दा माथिबाट तलतिर हेर्दा क्षैतिज रेखासँग बनेको कोणलाई अवनतांश कोण भनिन्छ।
The angle formed with the horizontal line when looking down at an object.
मुख्य सूत्रहरू (Key Formulas)
$$tan~\theta = \frac{उचाई}{दुरी}$$
$$sin~\theta = \frac{उचाई}{कर्ण}$$
$$cos~\theta = \frac{दुरी}{कर्ण}$$
उदाहरण (Example)
एउटा भवनको छतबाट नदीको किनारामा रहेको रुखको फेदमा अवनतांश कोण $30^{\circ}$ छ। यदि भवनको उचाई 50 मि. छ भने रुखको फेददेखि भवनको दुरी पत्ता लगाउनुहोस्।
Solution:
मानौं भवनको दुरी = $x$ मि.
$tan~30^{\circ} = \frac{50}{x}$
$\frac{1}{\sqrt{3}} = \frac{50}{x}$
$x = 50\sqrt{3}$ मि.
$x \approx 50 \times 1.732 = 86.6$ मि.
∴ भवनको दुरी = $86.6$ मि. (लगभग)
7. त्रिकोणमितीय मान तालिका (Trigonometric Values Table)
| कोण | $0^{\circ}$ | $30^{\circ}$ | $45^{\circ}$ | $60^{\circ}$ | $90^{\circ}$ | $120^{\circ}$ | $135^{\circ}$ | $150^{\circ}$ | $180^{\circ}$ |
|---|---|---|---|---|---|---|---|---|---|
| $sin$ | 0 | $\frac{1}{2}$ | $\frac{1}{\sqrt{2}}$ | $\frac{\sqrt{3}}{2}$ | 1 | $\frac{\sqrt{3}}{2}$ | $\frac{1}{\sqrt{2}}$ | $\frac{1}{2}$ | 0 |
| $cos$ | 1 | $\frac{\sqrt{3}}{2}$ | $\frac{1}{\sqrt{2}}$ | $\frac{1}{2}$ | 0 | $-\frac{1}{2}$ | $-\frac{1}{\sqrt{2}}$ | $-\frac{\sqrt{3}}{2}$ | -1 |
| $tan$ | 0 | $\frac{1}{\sqrt{3}}$ | 1 | $\sqrt{3}$ | $\infty$ | $-\sqrt{3}$ | -1 | $-\frac{1}{\sqrt{3}}$ | 0 |
| $cosec$ | $\infty$ | 2 | $\sqrt{2}$ | $\frac{2}{\sqrt{3}}$ | 1 | $\frac{2}{\sqrt{3}}$ | $\sqrt{2}$ | 2 | $\infty$ |
| $sec$ | 1 | $\frac{2}{\sqrt{3}}$ | $\sqrt{2}$ | 2 | $\infty$ | -2 | $-\sqrt{2}$ | $-\frac{2}{\sqrt{3}}$ | -1 |
| $cot$ | $\infty$ | $\sqrt{3}$ | 1 | $\frac{1}{\sqrt{3}}$ | 0 | $-\frac{1}{\sqrt{3}}$ | -1 | $-\sqrt{3}$ | $\infty$ |
| कोण | $210^{\circ}$ | $225^{\circ}$ | $240^{\circ}$ | $270^{\circ}$ | $300^{\circ}$ | $315^{\circ}$ | $330^{\circ}$ | $360^{\circ}$ |
|---|---|---|---|---|---|---|---|---|
| $sin$ | $-\frac{1}{2}$ | $-\frac{1}{\sqrt{2}}$ | $-\frac{\sqrt{3}}{2}$ | -1 | $-\frac{\sqrt{3}}{2}$ | $-\frac{1}{\sqrt{2}}$ | $-\frac{1}{2}$ | 0 |
| $cos$ | $-\frac{\sqrt{3}}{2}$ | $-\frac{1}{\sqrt{2}}$ | $-\frac{1}{2}$ | 0 | $\frac{1}{2}$ | $\frac{1}{\sqrt{2}}$ | $\frac{\sqrt{3}}{2}$ | 1 |
| $tan$ | $\frac{1}{\sqrt{3}}$ | 1 | $\sqrt{3}$ | $\infty$ | $-\sqrt{3}$ | -1 | $-\frac{1}{\sqrt{3}}$ | 0 |
महत्त्वपूर्ण नोट (Important Note):
कोणहरूको मान $90^{\circ} \pm \theta$, $180^{\circ} \pm \theta$, $270^{\circ} \pm \theta$, $360^{\circ} \pm \theta$ का लागि त्रिकोणमितीय अनुपातहरूको चिन्ह र मान परिवर्तन हुन्छ। यसलाई “ASTC” नियमले याद गर्न सकिन्छ।
8. Full Chapter PDF Manual
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