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In
DeltaUVW,w=5.3cm,v=4.8cm and
/_V=151^(@). Find all possible values of
/_W, to the nearest 1oth of a degree.
Answer:

In $\Delta \mathrm{UVW}, w=5.3 \mathrm{~cm}, v=4.8 \mathrm{~cm}$ and $\angle \mathrm{V}=151^{\circ}$ . Find all possible values of $\angle \mathrm{W}$ , to the nearest $10$ th of a degree. $\newline$ Answer:

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Evaluate variable expressions for number sequences

Full solution

Q. In

\Delta \mathrm{UVW}, w=5.3 \mathrm{~cm}, v=4.8 \mathrm{~cm}

and

\angle \mathrm{V}=151^{\circ}

. Find all possible values of

\angle \mathrm{W}

, to the nearest

10

th of a degree.

\newline

Answer:

Use Law of Sines: Use the Law of Sines to find the ratio of the sides to the sines of their opposite angles. $\newline$ $\frac{w}{\sin W} = \frac{v}{\sin V}$ $\newline$ Substitute the given values into the equation. $\newline$ $\frac{5.3}{\sin W} = \frac{4.8}{\sin 151^\circ}$
Calculate sine of V: Calculate the sine of angle V. $\newline$ $\sin 151^\circ \approx 0.5150$
Substitute and solve: Substitute the value of sin V into the equation and solve for sin W. $\newline$ $\frac{5.3}{\sin W} = \frac{4.8}{0.5150}$ $\newline$ $\sin W = \frac{5.3 \times 0.5150}{4.8}$ $\newline$ $\sin W \approx \frac{2.7295}{4.8}$ $\newline$ $\sin W \approx 0.5686$
Find possible values of W: Find the possible values of angle W using the inverse sine function. $\newline$ $W \approx \sin^{-1}(0.5686)$ $\newline$ Since the sine function has a range of [ $-1$ , $1$ ] and is positive in the first and second quadrants, there are two possible angles for W: one acute and one obtuse. $\newline$ $W_1 \approx \sin^{-1}(0.5686)$ $\newline$ $W_2 \approx 180^\circ - W_1$
Calculate first W: Calculate the first possible value of angle W. $\newline$ $W_1 \approx \sin^{-1}(0.5686)$ $\newline$ $W_1 \approx 34.7^\circ$
Calculate second W: Calculate the second possible value of angle W. $\newline$ $W_2 \approx 180^\circ - 34.7^\circ$ $\newline$ $W_2 \approx 145.3^\circ$
Check triangle angles: Check if the sum of angles in the triangle exceeds $180$ degrees when using the second possible value of angle W. $\newline$ $151^\circ + 145.3^\circ > 180^\circ$ $\newline$ This is not possible in a triangle, so the only valid solution for angle W is the acute angle.

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