Question

Solve these questions in your notebook and post ... Please fast (◍•ᴗ•◍)❤​

Answer 1

Question 1 :

If 3cot θ = 2 show that ( 4sin θ - 3cos θ ) / ( 2sin θ + 6cos θ ) = 1/3

Answer :

Given :

3cot θ = 2

⇒ cot θ = 2/3

Now let's take the equation that needs to be shown

$\sf \dfrac{4sin \theta - 3cos \theta }{2sin \theta + 6 cos \theta}=\dfrac{1}{3}$

Consider LHS

$\sf =\dfrac{4sin \theta - 3cos \theta }{2sin \theta + 6 cos \theta}$

Dividing both numerator and denominator by sin θ

$\sf =\dfrac{\dfrac{4sin \theta - 3cos \theta}{sin \theta} }{\dfrac{2sin \theta + 6 cos \theta}{sin \theta}} \\\\\\ \sf =\dfrac{\dfrac{4sin \theta}{sin \theta} - \dfrac{3cos \theta}{sin \theta } }{\dfrac{2sin \theta}{sin \theta } +\dfrac{ 6 cos \theta}{sin \theta}} \\\\\\ \sf =\dfrac{4 - \dfrac{3cos \theta}{sin \theta } }{2 +\dfrac{ 6 cos \theta}{sin \theta}}$

Since cos θ / sin θ

$\sf= \dfrac{4-3cot \theta}{2+6cot \theta}$

Substituting cot θ = 2/3 we get,

$\sf= \dfrac{4-3\bigg(\dfrac{2}{3}\bigg) }{2+6\bigg( \dfrac{2}{3} \bigg)} \\\\\\ \sf= \dfrac{4-2}{2+4} \\\\\\ \sf= \dfrac{2}{6} \\\\\\ \sf= \dfrac{1}{3} \\\\\\ \sf = RHS$

Hence Shown.

Question 2 :

Prove that : ( sin θ - 2sin³ θ ) / ( 2cos³ θ - cos θ ) = tan θ

Answer :

$\sf \dfrac{sin \theta - 2sin^3 \theta}{2cos^3 \theta -cos \theta} =tan \theta$

Consider LHS

$\sf= \dfrac{sin \theta - 2sin^3 \theta}{2cos^3 \theta -cos \theta}$

Taking sin θ common in numerator and cos θ in denominator

$\sf= \dfrac{sin \theta ( 1- 2sin^2 \theta)}{cos \theta ( 2cos^2 \theta - 1 )}$

Using trignometric identity cos² θ = 1 - sin² θ we get,

$\sf= \dfrac{sin \theta ( 1- 2sin^2 \theta)}{cos \theta [ 2(1-sin^2 \theta) - 1 ]} \\\\\\ \sf= \dfrac{sin \theta ( 1- 2sin^2 \theta)}{cos \theta ( 2-2sin^2 \theta - 1 )}\\\\\\ \sf= \dfrac{sin \theta ( 1- 2sin^2 \theta)}{cos \theta (1- 2sin^2 \theta )}\\\\\\ \sf = \dfrac{ sin \theta }{cos \theta }$

Since sin θ / cos θ = tan θ

= tan θ

= RHS

Hence proved.

Answer 2

Answer:

Basics. The present perfect tense refers to an action or state that either occurred at an indefinite time in the past (e.g., we have talked before) or began in the past and continued to the present time (e.g., he has grown impatient over the last hour). This tense is formed by have/has + the past participle.