Question

HELP ASAP TIMED !!!! What is the exact value of cos(c + d), given Sine c = StartFraction 24 Over 25 EndFraction for c in Quadrant II and Cosine d = negative three-fourths for d in Quadrant III?
Negative StartFraction 47 Over 100 EndFraction
Negative 1 and StartFraction 3 Over 100 EndFraction
StartFraction 21 minus 24 StartRoot 7 EndRoot Over 100 EndFraction
StartFraction 21 + 24 StartRoot 7 EndRoot Over 100 EndFraction

Answer 1

9514 1404 393

Answer:

  D

Step-by-step explanation:

In the second quadrant, ...

  c = 180° -arcsin(24/25) ≈ 106.26°

In the third quadrant, ...

  d = 360° -arccos(-3/4) ≈ 221.41°

Then cos(c+d) = cos(327.67°) ≈ 0.84498

This is a positive irrational number, greater than 21/100, so the only reasonable choice is the last one:

  $\dfrac{21+24\sqrt{7}}{100}\approx 0.84498$

_____

Perhaps you want to work this out using the trig identities.

  cos(c) = -√(1 -sin(c)²) = -7/25

  sin(d) = -√(1 -cos(d)²) = -(√7)/4

Then the desired cosine is ...

  cos(c+d) = cos(c)cos(d) -sin(c)sin(d)

  cos(c+d) = (-7/25)(-3/4) -(24/25)(-√7/4)

  cos(c+d) = (21 +24√7)/100 . . . . matches choice D