Question

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?

Answer 1

The exact value of cos(c+d) is (D) $cos(c+d) = \frac{21+24\sqrt{7} }{100}$.

What are trigonometric functions?

• Trigonometric functions are real functions in mathematics that connect an angle of a right-angled triangle to ratios of two side lengths.
• They are widely utilized in all geosciences, including navigation, solid mechanics, celestial mechanics, geodesy, and many more.

To find the exact value of cos(c+d):

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:

$\frac{21+24\sqrt{7} }{100}$ ≈ $0.84498$

To prove: Perhaps you want to work this out using the trignometric 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)

Therefore, the exact value of cos(c+d) is (D) $cos(c+d) = \frac{21+24\sqrt{7} }{100}$.

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The complete question is given below:

What is the exact value of cos(c+d), given sine c = start fraction 24 over 25 end fraction for c in quadrant ii and cosine d = negative three-fourths for d in quadrant iii?

(A) Negative StartFraction 47 Over 100 EndFraction

(B) Negative 1 and StartFraction 3 Over 100 EndFraction

(C) StartFraction 21 minus 24 StartRoot 7 EndRoot Over 100 EndFraction

(D) StartFraction 21 + 24 StartRoot 7 EndRoot Over 100 EndFraction