Question

if sin theta = m²- n² / m²+n², the find trigonometric ratios

Answer 1

Solution

→ sin∅ = (m² - n²)/(m² + n²)

Since sin∅ = Perpendicular/Hypotenuse.

→ H² = B² + P²

→ (m² + n²)² = B² + (m² - n²)²

→ B² = (m² + n²)² - (m² - n²)²

→ B² = (m²+n²+m²-n²)(m²+n²-m²+n²)

→ B = √[2m² × 2n²]

→ B = 2mn

• cos∅ = B/H
• sec∅ = H/B
• cosec∅ = H/P
• tan∅ = P/B
• cot∅ = B/P

→ cos∅ = 2mn/(m ²+ n²)

→ sec∅ = (m² + n²)/2mn

→ cosec∅ = (m² + n²)/(m² - n²)

→ tan∅ = (m² - n²)/2mn

→ cot∅ = 2mn/(m² - n²)

Answer 2

Answer:

$\sf \implies cos\theta = \dfrac{2mn}{(m^2+ n^2)}\\ \\\implies sec\theta = \dfrac{(m^2 + n^2)}{2mn}\\ \\\implies cosec \theta = \dfarc{(m^2 + n^2)}{(m^2- n^2)}\\ \\ \implies tan \theta = \dfrac{(m^2 - n^2)}{2mn}\\ \\ \implies cot \theta= \dfrac{2mn}{(m^2 - n^2)}$

Step-by-step explanation:

Given that :

$\sf Sin\theta = \dfrac{(m^2 - n^2)}{(m^2 + n^2)}$

To Find :

Trigonometric ratios.

Solution :

Let us know that what are trigonometric ratios.

• Cos∅  
• Sec∅
• Cosec∅  
• Tan∅  
• Cot∅

We know that :

Pythagoras theorem :

$\bigstar \boxed{\sf P^2 + B^2 = H^2}}}$

$\sf Sin\theta = \dfrac{Perpendicular.}{Hypotenuse.}$

So,

$\sf \implies (m^2 + n^2)^2 = B^2 + (m^2 - n^2)^2\\ \\ \implies B^2 = (m^2 + n^2)^2 - (m^2 - n^2)^2\\ \\\implies B^2 = (m^2 +n^2+m^2-n^2)(m^2+n^2 -m^2+n^2)\\ \\\implies B = \sqrt{2m^2 \times 2n^2}\\ \\ \implies B = 2mn$

$\textbf{\underline{\underline{Remember : - }}}\\\\\sf \longrightarrow cos\theta = \dfrac{B}{H}\\ \\ \longrightarrow sec\theta = \dfrac{H}{B}\\ \\ \longrightarrow cosec\theta = \dfrac{H}{P}\\ \\\longrightarrow tan\theta = \dfrac{P}{B}\\ \\\longrightarrow cot\theta = \dfrac{B}{P}$

We know that :

$\tt \longrightarrow B=2mn \\\\ \longrightarrow H = (m^2+ n^2)\\\\\longrightarrow P= (m^2 - n^2)$

After putting values :

$\sf \implies cos\theta = \dfrac{2mn}{(m^2+ n^2)}\\ \\\implies sec\theta = \dfrac{(m^2 + n^2)}{2mn}\\ \\\implies cosec \theta = \dfarc{(m^2 + n^2)}{(m^2- n^2)}\\ \\ \implies tan \theta = \dfrac{(m^2 - n^2)}{2mn}\\ \\ \implies cot \theta= \dfrac{2mn}{(m^2 - n^2)}$

$\rule{300}{1.5}$

★ Extra Information :

$\boxed{\begin{minipage}{7cm}{\underline{\underline{{\textbf {Fundamental Trignometric\:\: Indentities:}}}}}\\ \\ \sf \sin^{2}\theta+\cos^{2}\theta =1 \\ \\ 1+tan^{2}\theta=\sec^{2}\theta \\ \\ 1 + cot^{2}\theta=\text{cosec}^2\theta \\ \\ tan\theta =\dfrac{sin\theta}{cos\theta} \\ \\ cot\theta =\dfrac{cos\theta}{sin\theta} \end{minipage}}$