Question

Consider right triangle ∆ghi below. A right triangle ghi. Angle gih is a right angle. Angle ghi is seventy-five degrees. Side gh is eight point three units. Side gi is eight units.

Which expressions represent the length of side hi? Choose 2 answers:
a. 8.3⋅sin(90°-75°)
b. 8.3⋅cos(90°-75°)
c. 8⋅tan(90°-75°)
d. 8/tan(90°-75°)

Answer 1

In right triangle ∆ghi with a 75-degree angle ∠ghi, the correct expressions for the length of side hi are options (a) 8.3 * sin(90° - 75°) and (d) 8 / tan(90° - 75°).

Step-by-step explanation:

In right triangle ∆ghi, we have side gi (adjacent to ∠ghi) and side gh (hypotenuse), with ∠ghi equal to 75 degrees. To find the length of side hi, we need to use trigonometric ratios. Since we know the hypotenuse and an angle, and we want to find the opposite side, we can use the sine function. The sine of an angle in a right triangle is equal to the length of the opposite side over the hypotenuse (sin(∠) = opposite/hypotenuse).

Using this information, we calculate the length of side hi as follows:

Option (d): By trigonometric identities, tan(∠) = sin(∠) / cos(∠), we can express side hi as a function of the tangent: hi = gi * tan(∠ghi) = 8 * tan(75°). So side hi can also be found using the tangent ratio. If we take the reciprocal of tan(90° - 75°), which is cot(75°), it is equal to 1/tan(75°). So hi = gi / tan(75°) = 8 / tan(75°), making option (d) correct. Therefore, the correct expressions for the length of side hi are options (a) and (d).