Question

in triangle RST s=50 and angle T=45 degrees using simplified radicals when appropriate, find the range of values of t for which there are

a)2 possible measures for angle S
b)exactly 1 measure for angle S

Answer 1

In triangle RST $s = 50$ and angle $T = 45$ degrees using simplified radicals when appropriate, find the range of values of $t$ for which there are

a. 2 possible measures for angle $S$ ;

b. exactly 1 measure for angle $S$ .

Solution.

a. Let $S_{1}$ and $S_{2}$ are two possible measures for angle $S$ :

$$\angle S _ {1} < \angle S < \angle S _ {2}$$

Let the range of values of $t$ is $(t_1, t_2)$

$S_{1}$ and $S_{2}$ must be in the range $(0{}^{\circ}, 135{}^{\circ})$ .

Use law of sines to find the range of values of $t$ .

First possible measure:

$$\frac {t _ {1}}{\sin \angle T} = \frac {s}{\sin \angle S _ {1}}$$ $$t _ {1} = \frac {s \cdot \sin 4 5 {}^ {\circ}}{\sin \angle S _ {1}} = \frac {2 5 \sqrt {2}}{\sin \angle S _ {1}}$$

Second possible measure:

$$\frac {t _ {2}}{\sin \angle T} = \frac {s}{\sin \angle S _ {2}}$$ $$t _ {2} = \frac {2 5 \sqrt {2}}{\sin \angle S _ {2}}$$

Answer:

$$\frac {2 5 \sqrt {2}}{\sin \angle S _ {1}} < t < \frac {2 5 \sqrt {2}}{\sin \angle S _ {2}}$$

b.

Similarly use law of sines to find $t$ :

$$t = \frac {\frac {t}{\sin \angle T} = \frac {s}{\sin \angle S}}{s \cdot \sin \angle S} = \frac {5 0 \cdot \sqrt {2}}{2 \cdot \sin \angle S}$$

Answer: $t = \frac{50 \cdot \sqrt{2}}{2 \cdot \sin \angle S}$ .