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Question: If \( \sin x + \sin y + \sin z = 3 \) . Find the value of \( \cos x + \cos y + \cos z \) ....

If sin⁡x+sin⁡y+sin⁡z=3\sin x + \sin y + \sin z = 3 . Find the value of cos⁡x+cos⁡y+cos⁡z\cos x + \cos y + \cos z .

Explanation

Solution

Hint : We have been given the sum of sine angles, using its interval values, we can find which satisfies the equation and then find the value of respective angles. We can then substitute these values of angles in the sum of their required cosines to get the required answer.
The value of sin⁡θ\sin \theta lies between the interval [-1, 1]

Complete step-by-step answer :
It is given that sin⁡x+sin⁡y+sin⁡z=3\sin x + \sin y + \sin z = 3
We know that the value of sin⁡θ\sin \theta lies between the interval [-1, 1]. We can try substituting three values lying in this interval i.e. -1, 0 and 1 and see which of them satisfies the equation:
\-1−1−1≠3 0+0+0≠3 1+1+1=3   \- 1 - 1 - 1 \ne 3 \\\ 0 + 0 + 0 \ne 3 \\\ 1 + 1 + 1 = 3 \;
Thus, the given equation will be satisfied when the value of all the respective sine angles will be 1.
⇒sin⁡x=sin⁡y=sin⁡z=sin⁡θ=1 ⇒x=y=z=θ......(1)   \Rightarrow \sin x = \sin y = \sin z = \sin \theta = 1 \\\ \Rightarrow x = y = z = \theta ......(1) \;
Now, the value of sine is 1 at an angle of 90°. So the measure of the angles is 90°
⇒sin⁡θ=1 ⇒sin⁡θ=sin⁡90∘ ⇒θ=90∘   \Rightarrow \sin \theta = 1 \\\ \Rightarrow \sin \theta = \sin {90^\circ } \\\ \Rightarrow \theta = {90^\circ } \;
Then using (1), we get:
x=y=z=θ x=y=z=90∘....(2)   x = y = z = \theta \\\ x = y = z = {90^\circ }....(2) \;
According to the question, we are required to find the value of cos⁡x+cos⁡y+cos⁡z\cos x + \cos y + \cos z
Substituting the values of angles from (2), we get:
⇒cos⁡90∘+cos⁡90∘+cos⁡90∘ ⇒0+0+0=0(cos⁡90∘=0)   \Rightarrow \cos {90^\circ } + \cos {90^\circ } + \cos {90^\circ } \\\ \Rightarrow 0 + 0 + 0 = 0\left( {\cos {{90}^\circ } = 0} \right) \;
Therefore, the required value of cos⁡x+cos⁡y+cos⁡z\cos x + \cos y + \cos z is 0 when sin⁡x+sin⁡y+sin⁡z=3\sin x + \sin y + \sin z = 3

Note : In general, we know that when sine is 1 cosine will be 0, when sine is 12\dfrac{1}{2} cosine will be 32\dfrac{{\sqrt 3 }}{2} , etc. So when the value of sine of an angle is 1, we can directly say that the value of cosine for that angle will be 0 and have directly given the answer without finding the measure of the respective angles formed. Both sine and cosine are called complementary angles that means their sum is equal to 90°.