Math / Trigonometry

QuestionNote: Triangle may not be drawn to scale. Suppose $\mathrm{a}=11$ and $\mathrm{A}=25$ degrees. Find: $b=$ $\square$ $c=$ $\square$ $B=$ $\square$ degrees
Give all answers to at least one decimal place. Give angles in degrees

Studdy Solution

STEP 1

1. The triangle is a right triangle.
2. Angle $A$ is $25^\circ$ .
3. Side $a$ is opposite angle $A$ and is $11$ .
4. We need to find side $b$ (adjacent to angle $A$ ), side $c$ (the hypotenuse), and angle $B$ .

STEP 2

1. Use trigonometric ratios to find side $b$ .
2. Use trigonometric ratios to find side $c$ .
3. Use the angle sum property of triangles to find angle $B$ .

STEP 3

Use the tangent function to find side $b$ :
$\tan(A) = \frac{a}{b} \implies b = \frac{a}{\tan(A)}$
Substitute the given values:
$b = \frac{11}{\tan(25^\circ)}$
Calculate $b$ :
$b \approx \frac{11}{0.4663} \approx 23.6$

STEP 4

Use the sine function to find side $c$ :
$\sin(A) = \frac{a}{c} \implies c = \frac{a}{\sin(A)}$
Substitute the given values:
$c = \frac{11}{\sin(25^\circ)}$
Calculate $c$ :
$c \approx \frac{11}{0.4226} \approx 26.0$

STEP 5

Use the angle sum property of triangles to find angle $B$ :
$A + B + 90^\circ = 180^\circ \implies B = 180^\circ - 90^\circ - A$
Substitute the given value for $A$ :
$B = 180^\circ - 90^\circ - 25^\circ = 65^\circ$
The solutions are:
$b \approx 23.6, \quad c \approx 26.0, \quad B = 65^\circ$

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