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The exact value of cos(c+d) is (D) [tex]cos(c+d) = \frac{21+24\sqrt{7} }{100}[/tex].
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:
[tex]\frac{21+24\sqrt{7} }{100}[/tex] ≈ [tex]0.84498[/tex]
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) [tex]cos(c+d) = \frac{21+24\sqrt{7} }{100}[/tex].
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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
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