1. ## Finding The Angle In Radians

1. In a circle with radius 12 cm, an arc of length 20 cm subtends a central angle of theta. Determine the measure of theta in radians.

Answer: 1.67 (Rounded to two decimals)

My work:
(theta in radians) x 12 = 20
Unsure how to continue...

2. Solve: 7tan(x) = -3 0 <= x <= 2pi

(Rounded to two decimals)

My work:
tan(x) = -3/7

3. For the function f(x) = 3sin(bx) + d, where b and d are positive constants, determine an expression for the smallest positive value of x that produces the maximum value of f(x)

How and Why?

2. Originally Posted by AlphaRock
1. In a circle with radius 12 cm, an arc of length 20 cm subtends a central angle of theta. Determine the measure of theta in radians.

Answer: 1.67 (Rounded to two decimals)

My work:
(theta in radians) x 12 = 20
Unsure how to continue...
$\displaystyle \frac{20}{12} = \frac{5}{3} \approx 1.67$

2. Solve: 7tan(x) = -3 0 <= x <= 2pi

(Rounded to two decimals)

My work:
tan(x) = -3/7
$\displaystyle x = \tan^{-1}\left(-\frac{3}{7}\right)$
$\displaystyle x \approx -0.40$

You need answers in $\displaystyle 0 \le x \le 2\pi$, but when you take the inverse tangent, you get an answer in $\displaystyle -\pi/2 \le x \le \pi/2$ (because the latter is the range of the inverse tangent). Since our answer is negative, add $\displaystyle \pi$ to it to get $\displaystyle x \approx 2.74$. Then add $\displaystyle \pi$ to it again to get $\displaystyle x \approx 5.88$. Don't add $\displaystyle \pi$ a third time, because you'll then get an answer greater than $\displaystyle 2\pi$.

01

3. Originally Posted by AlphaRock
3. For the function f(x) = 3sin(bx) + d, where b and d are positive constants, determine an expression for the smallest positive value of x that produces the maximum value of f(x)

How and Why?
If you look at the basic sine function
$\displaystyle y = \sin x$,
the maximum value (for y) is 1, so the smallest positive value of x that gives the maximum value is
$\displaystyle x = \frac{\pi}{2}$.

$\displaystyle f(x) = 3\sin (bx) + d$,
since there is no reflection on the x-axis (ie. no negative in front of the 3), and since b and d are positive, the smallest positive value of x that gives the maximum value is when what's inside the sine is $\displaystyle \frac{\pi}{2}$, or
$\displaystyle bx = \frac{\pi}{2}$.
$\displaystyle x = \frac{\pi}{2b}$.