Solve the right triangle with the given parts. $\newline$ $\mathrm{A}=47.7^{\circ}, \mathrm{B}=42.3^{\circ}$ $\newline$ ielect the correct answer below and, if necessary, fill in the answer boxes to complete your choice. $\newline$ A. $a=$ $\square$ $\mathrm{b}=$ $\square$ $\newline$ $\square$ (Round to one decimal place as needed.) $\newline$ B. There is not enough information to solve the triangle.

View step-by-step help

Home

Math Problems

Algebra 1

Power rule with rational exponents

Full solution

Q. Solve the right triangle with the given parts.

\newline

\mathrm{A}=47.7^{\circ}, \mathrm{B}=42.3^{\circ}

\newline

ielect the correct answer below and, if necessary, fill in the answer boxes to complete your choice.

\newline

a=

\square

\mathrm{b}=

\square

\newline

\square

(Round to one decimal place as needed.)

\newline

B. There is not enough information to solve the triangle.

Find Angle C: Since we have a right triangle, we know that angle C is $90$ degrees. We can find angle C by subtracting the sum of angles A and B from $180$ degrees. $\newline$ $C = 180 - (A + B)$ $\newline$ $C = 180 - (47.7 + 42.3)$ $\newline$ $C = 180 - 90$ $\newline$ $C = 90$ degrees
Use Sine Function: Now we have confirmed that angle $C$ is indeed $90$ degrees, which makes sense for a right triangle. We can use the sine and cosine functions to find the lengths of sides $a$ and $b$ . Let's use the sine function for angle $A$ to find side $a$ . $\newline$ $\sin(A) = \frac{\text{opposite}}{\text{hypotenuse}}$ $\newline$ $\sin(47.7) = \frac{a}{c}$
Use Cosine Function: We don't have the length of the hypotenuse $c$ , so we can't directly calculate side $a$ yet. Let's use the cosine function for angle $A$ to find side $b$ . $\newline$ $\cos(A) = \frac{\text{adjacent}}{\text{hypotenuse}}$ $\newline$ $\cos(47.7) = \frac{b}{c}$
Find Side b: Again, we don't have the length of the hypotenuse $c$ , so we can't directly calculate side $b$ either. We need to use the sine function for angle $B$ to find side $b$ , since we know that side $b$ is the opposite side to angle $B$ . $\sin(B) = \frac{\text{opposite}}{\text{hypotenuse}}$ $\sin(42.3) = \frac{b}{c}$
Apply Pythagorean Theorem: We can use the Pythagorean theorem to find the hypotenuse $c$ , since we have a right triangle. $\newline$ $a^2 + b^2 = c^2$ $\newline$ But we don't have the lengths of sides $a$ and $b$ , so we can't find $c$ yet. We need to find one of the sides first.
Assume Hypotenuse: Let's assume the hypotenuse $c$ is $1$ for simplicity, which allows us to find the lengths of $a$ and $b$ relative to $c$ . $\newline$ $\sin(47.7) = \frac{a}{1}$ $\newline$ $a = \sin(47.7)$
Calculate Value of a: Now we calculate the value of $a$ using a calculator. $a \approx \sin(47.7 \text{ degrees})$ $a \approx 0.736$
Calculate Value of b: Next, we calculate the value of b using the sine of angle $B$ . $\newline$ $\sin(42.3) = \frac{b}{1}$ $\newline$ $b = \sin(42.3)$
Lengths of Sides: Now we calculate the value of $b$ using a calculator. $b \approx \sin(42.3 \text{ degrees})$ $b \approx 0.669$
Math Error: We have the lengths of sides $a$ and $b$ relative to the hypotenuse $c$ being $1$ . However, we assumed $c$ to be $1$ without any basis, which is incorrect. We cannot assume the hypotenuse to be $1$ without additional information. This is a math error.