186k views
1 vote
If cot x=v3, calculate (sin x)(cos x)-cos²x

User Aultimus
by
8.2k points

1 Answer

3 votes

Answer:


sin(x)\,cos(x)-cos^2(x)=(√(3) -3)/(4)

Explanation:

Recall the Pythagorean identity for cotangent:


1+cot^2(x)=csc^2(x)

where the cosecant is the reciprocal of the sine function. So, since we know the value of the cotangent of the angle, we can derive the value of the square of the sine:


1+cot^2(x)=csc^2(x)\\1+(√(3))^2=(1)/(sin^2(x)) \\1+3=(1)/(sin^2(x)) \\4=(1)/(sin^2(x))\\sin^2(x)=(1)/(4)\\ sin(x)=+/-(1)/(2)

We can also use the Pythagorean identity for sin and cos, to find the value of
cos^2(x) and of cos(x):


cos^2(x)=1-sin^2(x)\\cos^2(x)=1-(1)/(4) \\cos^2(x)=(3)/(4)\\cos(x)=+/-(√(3))/(2)

We also notice that since the cotangent is positive, the angle "x" must be located in either the firs or the third quadrant, where both sine and cosine have the same sign (both positive in the first quadrant, and both negative in the third quadrant.

Then the requested quantity can be written as:


sin(x)\,cos(x)-cos^2(x)=((1)/(2)) \,(√(3) )/(2) -(3)/(4) =(√(3) )/(4) -(3)/(4)=(√(3)-3 )/(4)

User ZippyV
by
8.1k points

Related questions

1 answer
2 votes
120k views
asked Sep 27, 2024 109k views
Ulix asked Sep 27, 2024
by Ulix
8.1k points
1 answer
2 votes
109k views
1 answer
5 votes
84.5k views