Smaller triangle: tan(22°) = opposite/10, so opposite = 10 tan(22°). This side is also the base of the larger triangle (let it be x). In the larger triangle, sin(22°) = x/m, so m = x/sin(22°). Substituting x, we get m = 10/cos(22°). Hence, m is 10 divided by the cosine of 22 degrees.
Identify the relevant triangles and angles.
We see two right triangles: the smaller triangle with hypotenuse 10 and angle 22°, and the larger triangle with hypotenuse m and angle 22°.
Apply trigonometric ratios to the smaller triangle.
Since we have the hypotenuse (10) and an acute angle (22°), we can use the tangent function (tan) to find the side opposite the 22° angle.
tan(22°) = (opposite side) / 10
opposite side = 10 * tan(22°)
Relate the smaller and larger triangles.
The side opposite the 22° angle in the smaller triangle is also the base of the larger triangle. Let's call this side "x".
Apply trigonometric ratios to the larger triangle.
We know the base (x) and need to solve for the hypotenuse (m). We also have the same angle (22°) as in the smaller triangle.
Since we have the base and an acute angle, we can use the sine function (sin) to find the hypotenuse.
sin(22°) = x / m
m = x / sin(22°)
Substitute and solve.
Substitute the expression for x from step 3 into the equation for m from step 4.
m = (10 * tan(22°)) / sin(22°)
Simplify the expression using the trigonometric identity: tan(22°) = sin(22°) / cos(22°).
m = 10 / cos(22°)
Therefore, the value of m is 10 / cos(22°). This can be further calculated using a calculator to obtain a numerical answer (approximately 11.54).