If cosX = 0.6 and sin X < 0, find cos(X + 3pi/4)?

Question

I have no idea on how the book came up with -7square root 2/10. I need a mathmatician to show their work on how this answer came up.

Answer 1

Ok...

cosx = 0.6
This means that x = arccos(0.6) = 53.13° or 306.87°.
But sinx < 0...
sin(53.13°) = 0.8
sin(306.87°) = -0.8
So the only possible value of x (within 0-360°) is 306.87°. 3pi/4 is 135°.

cos(306.87° + 135°) = cos(441.87°) = cos(81.87°) = 0.1414198
-7sqrt(2)/10 = -0.989949

What can I say; the numbers check... 306.87° satisfies the input conditions. -7sqrt(2)/10 is not the correct answer.

Answer 2

There might be a simpler way to solve this, but this is the only way I know how to solve it.

Since cos(x) = 0.6, cos(x) is a positive number. Therefore, the solution must be in quadrants 1 or 4.

Since sin(x) < 0, then sinx is negative, and sin is negative in quadrants 3 and 4.

Therefore, the solution must lie in quadrant 4

Recall the identity cos(A + B) = cosAcosB - sinAsinB. Let's plug that in for cos(x + 3pi/4)

cos(x + 3pi/4) =

cos(x)cos(3pi/4) - sin(x)sin(3pi/4)

Now, substitute normally.

(0.6)[-sqrt(2)/2] - sin(x)[sqrt(2)/2]

If cosx = 0.6, then cosx = 3/5. Remember the SOHCAHTOA rules; sin = opp/hyp, cos = adj/hyp, and tan = opp/adj

Since cos(x) = 3/5, make a right angle triangle, declaring x as an angle and make the adjacent side equal to 3 and the hypotenuse equal to 5. That means the opposite side is equal to 4, using the Pythagoreas theorem.

Therefore, sin(x) = opp/hyp = 4/5

So this: (0.6)[-sqrt(2)/2] - sin(x)[sqrt(2)/2]
Becomes this:
(0.6)[-sqrt(2)/2] - (4/5)[sqrt(2)/2]
-3sqrt(2)/10 - 2sqrt(2)/5
-3sqrt(2)/10 - 4sqrt(2)/10
-7sqrt(2)/10

Answer 3

If cos X = 0.6, it's a 3-4-5 triangle (0.6 = 3/5), sin X = -4/5 = -0.8 (it was given that sin X < 0)

cos (A + B) = cos A cos B - sin A sin B
= 0.6 cos (3pi/4) + 0.8 sin (3pi/4)
= 0.6 (-root(2)/2) - 0.8 root(2)/2
= -1.4 root(2)/2
= -0.7 root(2)
= -7 root(2)/10

Answer 4

2 pieces to start:

Cos(x) ^2 + Sin(x) ^2 = 1 so, sin(x) = sqrt(1 - cos(x) ^2) or

sin(x) = sqrt(1 - .36) = sqrt(.64) or +/- .8

They tell you that sin(x) < 0 so sin(x) = -0.8

Now, part 2 uses the double angle formula:
Cos(a + b) = cos(a)cos(b) - sin(a)sin(b)

so

cos(x + 3pi/4) = cos(x)cos(3pi/4) - sin(x)sin(3pi/4)

which gives: 0.6 * -sqrt(2)/2 - (-0.8) * sqrt(2)/2

I get 0.2 * sqrt(2)/2 or, rationalizing gives sqrt(2)/10

I'm assuming that the person who said the answer is:
-7 sqrt(2)/10

must have gotten that from -0.6 sqrt(2)/2 and -0.8 sqrt(2)/2 which would give -1.4 sqrt(2)/2 which would rationalize to:
-7sqrt(2)/10

BTW, the guy above that says that sin(x) = .8 isn't using the given that sin(x) < 0, therefore sin(x) = -0.8