Question

Let cos x = .28 and cos y = .74. Which is x – y?

A. cos^(–1).28 + cos^(–1).74
B. –cos^(–1).28 – cos^(–1).74
C. cos^(–1).28 – cos^(–1).74
D. –cos^(–1).28 + cos&(–1).74

Answer 1

Let $\cos x = .28$ and $\cos y = .74$. Which is $x - y$?

A. $\cos^{-1}0.28 + \cos^{-1}0.74$

B. $-\cos^{-1}0.28 - \cos^{-1}0.74$

C. $\cos^{-1}0.28 - \cos^{-1}0.74$

D. $-\cos^{-1}0.28 + \cos \text{amp}; (-1).74$

**Solution:**

Let us solve each equation:

$$\cos x = 0.28$$ $$x = \pm \cos^{-1} 0.28 + 2\pi k, \quad k - \text{integer constant}$$ $$\cos y = 0.74$$ $$y = \pm \cos^{-1} 0.74 + 2\pi n, \quad n - \text{integer constant}$$

Now we can simply find the difference between $x$ and $y$:

1. Angles in the range $0 < x < \pi, 0 < y < \pi$:

$$x - y = \cos^{-1} 0.28 - \cos^{-1} 0.74$$

Correct answer in this case is C

2. For all angles:

Since trigonometric equation has not one solution, all responses are correct because of the ambiguity of the angle (it may be $x = \cos^{-1} 0.28, -\cos^{-1} 0.28, \cos^{-1} 0.28 + 2\pi k, \ldots$,

$$y = \cos^{-1} 0.74, -\cos^{-1} 0.74, \cos^{-1} 0.28 + 2\pi k, \ldots)$$ $$x - y = \cos^{-1} 0.28 - (-\cos^{-1} 0.74) = \cos^{-1} 0.28 + \cos^{-1} 0.74 \text{ (answer A)}$$ $$x - y = -\cos^{-1} 0.28 - (\cos^{-1} 0.74) = -\cos^{-1} 0.28 - \cos^{-1} 0.74 \text{ (answer B)}$$ $$x - y = \cos^{-1} 0.28 - (\cos^{-1} 0.74) = \cos^{-1} 0.28 - \cos^{-1} 0.74 \text{ (answer C)}$$ $$x - y = -\cos^{-1} 0.28 - (-\cos^{-1} 0.74) = -\cos^{-1} 0.28 + \cos^{-1} 0.74 \text{ (answer D)}$$

**Answer:** Angles in the range $0 < x < \pi, 0 < y < \pi$: correct answer: C

For all angles: all answers are correct: A, B, C, D,

For the correct solution of the problem we need to know possible values of the angles $x$ and $y$.