If a -b =c, then a +b = - Vidyadip

MCQ

If $\text{a}\cos\theta-\text{b}\sin\theta=\text{c},$ then $\text{a}\sin\theta+\text{b}\cos\theta=$

A
$\pm\sqrt{\text{a}^2+\text{b}^2+\text{c}^2}$
✓
$\pm\sqrt{\text{a}^2+\text{b}^2-\text{c}^2}$
C
$\pm\sqrt{\text{c}^2-\text{a}^2-\text{b}^2}$
D
None of these.

✓

Answer

Correct option: B.

$\pm\sqrt{\text{a}^2+\text{b}^2-\text{c}^2}$

$\text{a}\cos\theta-\text{b}\sin\theta=\text{c}$
Squaring,
$\text{a}^2\cos^2\theta+\text{b}^2\sin^2\theta-2\text{ab}\sin\theta\cos\theta=\text{c}^2$
$\Rightarrow\ \text{a}^2(1-\sin^2\theta)+\text{b}^2(1-\cos^2\theta)-2\text{ab}\sin\theta\cos\theta=\text{c}^2$
$\Rightarrow\ \text{a}^2-\text{a}^2\sin^2\theta+\text{b}^2(1-\cos^2\theta)-2\text{ab}\sin\theta\cos\theta=\text{c}^2$
$\Rightarrow\ -\text{a}^2\sin^2\theta-\text{b}^2\cos^2\theta-2\text{ab}\sin\theta\cos\theta=\text{c}^2-\text{a}^2-\text{b}^2$
$\Rightarrow\ \text{a}^2\sin^2\theta+\text{b}^2\cos\theta+2\text{ab}\sin\theta\cos\theta=\text{a}^2+\text{b}^2-\text{c}^2 $ $\ (\text{Dividing by}-1)$
$(\text{a}\sin\theta+\text{b}\cos\theta)^2=\text{a}^2+\text{b}^2-\text{c}^2$
$\therefore\ \text{a}\sin\theta+\text{b}\cos\theta=\pm\sqrt{\text{a}^2+\text{b}^2-\text{c}^2}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "M.C.Q (1 Marks)" from Trigonometric Identities→Trigonometric Identities — all question types→Maths STD 10 question papers→CBSE Board STD 10 question papers→CBSE Board question paper generator→

Similar questions

The area of incircle of an equilateral triangle is $154\ cm^2$. The perimeter of the triangle is :

A pole stands vertically, inside a triangular park $\text{ABC}$. If the angle of elevation of the top of the pole from each corner of the park is same, then the foot of the pole is at the :

A box contains 3 blue, 2 white and 4 red marbles. If a marble is drawn at random from the box, what is the probability that it will not be a white marble?

The roots of the quadratic equation $x^2+ x -1=0$ are

To draw a pair of tangents to a circle, which are inclined to each other at angle of 45º, we have to draw the tangents at the end points of those two radii, the angle between which is:

The angles of elevation of the top of $12m$ high tower from two points in opposite directions with it are complementary. If distance of one point from its base is $16m,$ then distance of second point from tower's base is?

If $x$ is a positive integer such that the distance between points $P(x, 2)$ and $Q(3, -6)$ is $10$ units, then $x =$

The degree of a constant polynomial is :

A circular park has a path of uniform width around it. The difference betweenthe outer and inner circumferences of the circular path is $132\ m$. Its width is :

A train travels $360\ km$ at a uniform speed. If the speed had been $5\ km/ hr$ more, it would have taken $1$ hour less for the same journey, then the actual speed of the train is :