If `s intheta+sin^2theta1=1,` then prove that `cos^(12)theta+3cos^8theta+cos^6theta-1=0` Given that `s i lntheta=1-sin^2theta=1-sin^2theta=cos^2theta`
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If `cos ^(-1) .(x)/(a)+ cos ^(-1). (y)/(b) = theta` then prove that `(x^(2))/(a^(2)) -(2xy)/(ab) . cos theta+ (y^(2))/(b^(2)) = sin ^(2) theta`.
`cos^(-1).(x)/(a) + cos theta +(y^(2))/(b^(2)) = sin^(2) theta` `rArr cos^(-1)[(x)/(a).(y)/(b)-sqrt(1-(x^(2))/(a^(2)))sqrt(1-(y^(2))/(b^(2)))] = theta` `rArr (xy)/(ab) -sqrt(1-(x^(2))/(a^(2))-(y^(2))/(b^(2))+(x^(2)y^(2))/(a^(2)b^(2)))= cos theta` `rArr " " (xy)/(ab)-costheta =sqrt(1-(x^(2))/(a^(2))-(y^(2))/(b^(2))+(x^(3)y^(2))/(a^(2)b^(2)))` squaring both sideswe get `(x^(2)y^(2))/(a^(2)b^(2))+ cos^(2) theta -(2xy)/(ab)...
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Let `A=[{:(,sin theta,1sqrt2),(,-1//sqrt2,cos theta),(,cos theta,tan theta):}]& B=[{:(,1sqrt2,sin theta),(,cos theta,cos theta),(,cos theta,-1):}]`. F
By defing A & B are equal if they have the same order and all the corresponding elements are equal. Thus we have `sin theta=(1)/(sqrt2),c os theta=-(1)/(sqrt2)& tan theta=-1` `Rightarrow...
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Prove the orthogonal matrices of order two are of the form `[(cos theta,-sin theta),(sin theta,cos theta)]` or `[(cos theta,sin theta),(sin theta,-cos
Let matrix `A=[(a,b),(c,d)]` is orthogonal matrix. `:. [(a,b),(c,d)][(a,c),(b,d)]=[(1,0),(0,1)]` `implies [(a^(2)+b^(2),ac+bd),(ac+bd,c^(2)+d^(2))]=[(1,0),(0,1)]` `implies a^(2)+b^(2)=1` (1) `c^(2)+d^(2)=1` (2) `ac+bd=0` (3) `implies a/d= (-b)/c= k` (let) `implies c^(2)+d^(2)=(a^(2)+b^(2))/k=1//k^(2)` or `k^(2)=1` or `k= pm 1`...
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Find the characteristic roots of the two-rowed orthogonal matrix `[(cos theta,-sin theta),(sin theta,cos theta)]` and verify that they are of unit mod
We have, `|A-lambdaI|=|(cos theta - lambda,-sin theta),(sin theta,cos theta - lambda)|` `=(cos theta-lambda)^(2)+sin^(2) theta` Therefore, characteristic equation of A is `(cos theta-lambda)^(2)+sin^(2) theta=0` or `cos theta-lambda= pm i sin theta`...
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If `theta-phi=pi/2,` prove that, `[(cos^2 theta,cos theta sin theta),(cos theta sin theta,sin^2 theta)] [(cos^2 phi,cos phi sin phi),(cos phi sin phi,
Correct Answer - C `AB=[(cos^(2) theta,),(cos theta sin theta,)][(cos^(2) phi,cos phi sin phi),(cos phi sin phi,sin^(2) phi)]` `=[(cos^(2) theta cos^(2) phi+cos theta cos phi sin theta sin phi ,cos^(2)theta cos phi...
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If `A(theta)=[(sin theta, i cos theta),(i cos theta, sin theta)]`, then which of the following is not true ?
Correct Answer - A::B::C We have, `|A(theta)|=1` Hence, A is invertiable. `A(pi+theta)=A(pi + theta)=[(sin(pi+theta),i cos (pi+theta)),(i cos (pi+theta),sin (pi+theta))]` `=[(-sin theta,-i co theta),(-i cos theta,-sin theta)]=-A(theta)` adj `(A(theta))=[(sin theta,-i cos theta),(-i...
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If equation `sin^(-1) (4 sin^(20 theta + sin theta) + cos^(-1) (6 sin theta - 1) = (pi)/(2)` has 10 solution for `theta in [0, n pi]`, then find the m
Correct Answer - 9 We must have `4 sin^(2) theta + sin theta = 6 sin theta -1` `rArr 4 sin^(2) theta - 5 sin theta + 1 = 0` `rArr...
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सरल कीजिए, ` cos theta [{:( cos theta,, sin theta ), (-sin theta,, cos theta ):}] + sin theta [ {:(sin theta ,, - cos theta ), (cos theta ,, sin theta
Correct Answer - `[{:(,1,0),(,1,1):}]`
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Range of `f(theta)=cos^2theta(cos^2theta+1)+2 sin^2theta` is
Correct Answer - D `f(theta)=cos^2theta(cos^2theta+1)+2sin^2theta` `=cos^4theta+cos^2theta+sin^2theta+sin^2theta` `=cos^4theta+1+1-cos^2theta` `=(cos^2theta-1/2)^2+2-1/4` `=(cos^2theta-1/2)^2+7/4` `f_("min")=7/4` `f_("max")=1/4+7/4=2`
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If `abs(cos theta{sin theta+sqrt(sin^2theta+sin^2alpha)})lek`, then the value of k
Correct Answer - B Let `u=costheta{sintheta+sqrt(sin^2theta+sin^2alpha}` `or (u-sinthetacostheta)^2=cos^2theta(sin^2theta+sin^2alpha)` `or u^2tan^2theta-2utantheta+u^2-sin^2alpha=0` Since `tantheta` is real, we have `4u^2-4u^2(u^2-sin^2LPH)ge0` `or u^2le1+sin^2alpha` `or absulesqrt(1+sin^2alpha)`
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