Advertisements
Advertisements
Question
If`( 2 sin theta + 3 cos theta) =2 , " prove that " (3 sin theta - 2 cos theta) = +- 3.`
Advertisements
Solution
Given ,`(2 sin theta + 3 cos theta ) = 2 .....(i)`
We have `( 2 sintheta + 3 cos theta )^2 + ( 3 sin theta - 2 cos theta )^2`
=` 4 sin^2 theta + 9 cos^2 theta + 12 sin theta cos theta + 9 sin^2 theta + 4 cos^2 theta - 12 sin theta cos theta`
=`4 ( sin^2 theta + cos^2 theta ) + 9 ( sin^2 theta + cos^2 theta )`
=`4+9`
=13
i.e .,`( 2 sin theta + 3 cos theta ) ^2 + ( 3 sin theta - 2cos theta )^2 = 13`
= > `2^2 + (3 sintheta - 2 cos theta )^2 = 13`
= > `( 3 sin theta - 2 cos theta ) ^2 = 13-4`
= > `( 3 sin theta - 2 cos theta ) ^2 = 9 `
= > `( 3 sin theta - 2 cos theta ) = +- 3`
APPEARS IN
RELATED QUESTIONS
Prove the following trigonometric identities.
`(cos^2 theta)/sin theta - cosec theta + sin theta = 0`
Prove the following trigonometric identities.
if x = a cos^3 theta, y = b sin^3 theta` " prove that " `(x/a)^(2/3) + (y/b)^(2/3) = 1`
If 3 sin θ + 5 cos θ = 5, prove that 5 sin θ – 3 cos θ = ± 3.
Prove the following identities:
cot2 A – cos2 A = cos2 A . cot2 A
Show that : `sinA/sin(90^circ - A) + cosA/cos(90^circ - A) = sec A cosec A`
Show that : `sinAcosA - (sinAcos(90^circ - A)cosA)/sec(90^circ - A) - (cosAsin(90^circ - A)sinA)/(cosec(90^circ - A)) = 0`
(i)` (1-cos^2 theta )cosec^2theta = 1`
`(1+ tan theta + cot theta )(sintheta - cos theta) = ((sec theta)/ (cosec^2 theta)-( cosec theta)/(sec^2 theta))`
If `( cos theta + sin theta) = sqrt(2) sin theta , " prove that " ( sin theta - cos theta ) = sqrt(2) cos theta`
What is the value of \[\frac{\tan^2 \theta - \sec^2 \theta}{\cot^2 \theta - {cosec}^2 \theta}\]
If \[\sin \theta = \frac{1}{3}\] then find the value of 9tan2 θ + 9.
Prove that:
(cosec θ - sinθ )(secθ - cosθ ) ( tanθ +cot θ) =1
Prove the following identity :
`sqrt((1 + sinq)/(1 - sinq)) + sqrt((1- sinq)/(1 + sinq))` = 2secq
Find the value of `θ(0^circ < θ < 90^circ)` if :
`tan35^circ cot(90^circ - θ) = 1`
Without using trigonometric identity , show that :
`sin42^circ sec48^circ + cos42^circ cosec48^circ = 2`
Evaluate:
`(tan 65^circ)/(cot 25^circ)`
Prove the following identities.
cot θ + tan θ = sec θ cosec θ
To prove cot θ + tan θ = cosec θ × sec θ, complete the activity given below.
Activity:
L.H.S = `square`
= `square/sintheta + sintheta/costheta`
= `(cos^2theta + sin^2theta)/square`
= `1/(sintheta*costheta)` ......`[cos^2theta + sin^2theta = square]`
= `1/sintheta xx 1/square`
= `square`
= R.H.S
Prove that cot2θ – tan2θ = cosec2θ – sec2θ
Prove the following identity:
(sin2θ – 1)(tan2θ + 1) + 1 = 0
