if sin (-a)= - sin(a) why cos(-a) = cos(a)

Question

Question : Which of the following options gives an expression equivalent to $\sin \ (A + B)$?
Option 1: $\cos\ A \cos\ B - \sin \ A \sin \ B$
Option 2: $\sin\ A \cos \ B + \cos\ A\ \sin \ B$
Option 3: $\cos \ A \cos\ B+\sin\ A \sin \ B$
Option 4: $\sin \ A \cos \ B - \cos \ A \sin \ B$

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Answer 1

Hello Candidate,

Let me tell you that, there are 4 quadrants in cartesian plane.

In the first quadrant the angle is from 0 to 90, and here all the Trignometric functions are positive. In the 2nd quadrant, only Sinx and Cosecx are positive.

In 3rd quadrant only Tanx and Cotx are positive, whereas in 4th quadrant only Cosx and Secx are positive.

Now Sin(-x) means , Sin(360-x) , this value lies in the 4th quadrant and hence this becames -Sinx.

But for Cos(-x) , this is same Cos(360-x), which also lies in 4th quadrant. But we know that in 4th quadrant, Cosx is positive. Hence answer comes Cosx

Hope it helps.

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if sin (-a)= - sin(a) why cos(-a) = cos(a)

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Question : Which of the following options gives an expression equivalent to $\sin \ (A + B)$?
Option 1: $\cos\ A \cos\ B - \sin \ A \sin \ B$
Option 2: $\sin\ A \cos \ B + \cos\ A\ \sin \ B$
Option 3: $\cos \ A \cos\ B+\sin\ A \sin \ B$
Option 4: $\sin \ A \cos \ B - \cos \ A \sin \ B$

Question : Solve the following to find its value in terms of trigonometric ratios.
$(\sin A + \cos A)(1 - \sin A \cos A)$
Option 1: $\sin^3A+\cos^3A$
Option 2: $\sin^2A-\cos^2A$
Option 3: ${[\cos A-\sin A]\left[\sin ^2 A+\cos ^2 A\right]}$
Option 4: $\sin^3A-\cos^3A$

Question : If $\cos A+\sin A=\sqrt{2}\cos A$, then $\cos A-\sin A$ is equal to: (where $0^{\circ}< A< 90^{\circ}$)
Option 1: $\sqrt{2}\sin A$
Option 2: $2\sin A$
Option 3: $2\sqrt{\sin A}$
Option 4: $\sqrt{2\sin A}$