1. Solve the inequality:
Given: $-4 \leq \frac{x + 5}{3} \leq 7$
Multiply all parts by 3:
$-12 \leq x + 5 \leq 21$
Subtract 5 from all parts:
$-17 \leq x \leq 16$
Answer: $[-17, 16]$
2. Simplify $\cos(\theta) + \sin(\theta)\tan(\theta)$:
Recall $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$, so:
$\cos(\theta) + \sin(\theta) \cdot \frac{\sin(\theta)}{\cos(\theta)} = \cos(\theta) + \frac{\sin^2(\theta)}{\cos(\theta)}$
Combine terms:
$\frac{\cos^2(\theta) + \sin^2(\theta)}{\cos(\theta)} = \frac{1}{\cos(\theta)} = \sec(\theta)$
Answer: $\sec(\theta)$
3. Convert to polar form: $x^2 + (y - 6)^2 = 35$
Use substitutions: $x = r\cos(\theta)$, $y = r\sin(\theta)$
Substitute:
$(r\cos(\theta))^2 + (r\sin(\theta) - 6)^2 = 35$
Expand:
$r^2\cos^2(\theta) + r^2\sin^2(\theta) - 12r\sin(\theta) + 36 = 35$
Simplify using $\cos^2(\theta) + \sin^2(\theta) = 1$:
$r^2 - 12r\sin(\theta) + 36 = 35$
Rearrange:
$r^2 - 12r\sin(\theta) + 1 = 0$
Answer: $r^2 - 12r\sin(\theta) + 1 = 0$
Parent Tip: Review the logic above to help your child master the concept of 12th grade math worksheets.
