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$In /_\DEF,f=83cm,e=60cm and /_E=29^(@). Find all possible values of /_F, to the nearest 1oth of a degree. Answer:$

In $\triangle \mathrm{DEF}, f=83 \mathrm{~cm}, e=60 \mathrm{~cm}$ and $\angle \mathrm{E}=29^{\circ}$ . Find all possible values of $\angle \mathrm{F}$ , to the nearest $10$ th of a degree. $\newline$ Answer:

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Solve exponential equations by rewriting the base

Full solution

Q. In

\triangle \mathrm{DEF}, f=83 \mathrm{~cm}, e=60 \mathrm{~cm}

and

\angle \mathrm{E}=29^{\circ}

. Find all possible values of

\angle \mathrm{F}

, to the nearest

10

th of a degree.

\newline

Answer:

Calculate $\sin(E)$ : We can use the Law of Sines to find the possible values of angle F. The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. That is, for triangle DEF: $\newline$ $\frac{\sin(E)}{e} = \frac{\sin(F)}{f}$ $\newline$ First, we need to calculate $\sin(E)$ . $\newline$ $\sin(E) = \sin(29 \text{ degrees})$
Plug values into formula: Now we plug the values into the Law of Sines formula to find $\sin(F)$ . $\frac{\sin(F)}{f} = \frac{\sin(E)}{e}$ $\frac{\sin(F)}{83} = \frac{\sin(29 \text{ degrees})}{60}$
Solve for $\sin(F)$ : We solve for $\sin(F)$ . $\sin(F) = \left(\frac{\sin(29^\circ)}{60}\right) \times 83$
Calculate $\sin(F)$ : We calculate the value of $\sin(F)$ . $\sin(F) = \left(\frac{\sin(29^\circ)}{60}\right) * 83$ $\sin(F) \approx \left(\frac{0.4848}{60}\right) * 83$ $\sin(F) \approx 0.6706$
Correct calculation error: Since the sine of an angle cannot be greater than $1$ , we must have made a calculation error in the previous step. Let's correct it. $\newline$ $\sin(F) = \frac{\sin(29^\circ)}{60} \times 83$ $\newline$ $\sin(F) \approx \frac{0.4848}{60} \times 83$ $\newline$ $\sin(F) \approx 0.6706$ $\newline$ This is incorrect because $\sin(29^\circ) \approx 0.4848$ is already less than $1$ , so multiplying it by a fraction $\frac{83}{60}$ should give us a number less than $0.4848$ .

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