Question

If $5^\circ \leq x^\circ \leq 15^\circ$, then the value of $\sin 30^\circ + \cos x^\circ - \sin x^\circ$ will be:

• A. Between -1 and -0.5 inclusive
• B. Between -0.5 and 0 inclusive
• C. Between 0 and 0.5 exclusive
• D. Between 0.5 and 1 inclusive
• E. None of the above

Hint:

When solving trigonometric range problems, test the extreme values of $x$ within the interval to capture the possible range of the expression. Numerical approximation often quickly confirms the correct option.

Answer 1

Answer: (E)

Step 1: Simplify the given expression. \[ E = \sin 30^\circ + \cos x - \sin x \] Since $\sin 30^\circ = \tfrac{1}{2}$, \[ E = \tfrac{1}{2} + \cos x - \sin x \]

Step 2: Consider the range of $x$. $5^\circ \leq x \leq 15^\circ$. So $\cos x$ is close to 1, while $\sin x$ is small (between $\sin 5^\circ \approx 0.087$ and $\sin 15^\circ \approx 0.259$).

Step 3: Estimate bounds. - At $x=5^\circ$: \[ E = 0.5 + \cos 5^\circ - \sin 5^\circ \approx 0.5 + 0.9962 - 0.0872 = 1.409 \] - At $x=15^\circ$: \[ E = 0.5 + \cos 15^\circ - \sin 15^\circ \approx 0.5 + 0.9659 - 0.2588 = 1.207 \] So the value of $E$ lies in the interval: \[ 1.207 \leq E \leq 1.409 \]

Step 4: Compare with given options. - (A) $[-1,-0.5]$: Not valid. - (B) $[-0.5,0]$: Not valid. - (C) $(0,0.5)$: Not valid. - (D) $[0.5,1]$: Not valid (our values are above 1). - (E) None of the above: Correct.

Final Answer: \[ \boxed{\text{E. None of the above}} \]