Question

Someone please answer these 4 mathematical questions please

Answer 1

(5): Missing side: x = 13.9

(6): Missing side: x = 8.5

(7): Measure of indicated angle: ? = 46°

(8): Measure of indicated angle: ? = 35°

Explanation:

Because all four triangles are right triangles, we're able to find the side lengths and angles using trigonometry.

(5): When the 44° is the reference angle:

• The side that is 10 units long is the adjacent side,
• and the side that is x units long is the hypotenuse (side always opposite the right angle).

Thus, we can find x using the cosine ratio, which is given by:

cos (θ) = adjacent / hypotenuse, where

• θ is the measure of the reference angle.

Thus, we plug in 44 for θ, 10 for the adjacent side, and x for the hypotenuse and solve for x:

(cos (44) = 10 / x) * x

(x * cos (44) = 10) / cos (44)

x = 13.90163591

x = 13.9

Thus, x is about 13.9 units.

(6): When the 23° angle is the reference angle:

• The side that is x units long is the opposite side,
• and the side that is 20 units long is the adjacent side.

Thus, we can find x using the tangent ratio, which is given by:

tan (θ) = opposite / adjacent, where

• θ is the reference angle.

Thus, we plug in 23 for θ, x for the opposite side, and 20 for the adjacent side and solve for x:

(tan (23) = x / 20) * 20

8.489496324 = x

8.5 = x

Thus, x is about 8.5 units.

Since problems (7) and (8) require to find angles in a right triangle, we will need to use inverse trigonometry.

(7): When the unknown (?) angle is the reference angle:

• the side that is 25 units long is the opposite side,
• and the side that is 35 units long is the hypotenuse.

Thus, we can find the measure of the unknown (?) angle in ° using the inverse sine ratio which is given by:

sin^-1 (opposite / hypotenuse) = θ, where

• θ reference angle

Thus, we plug in 25 for the opposite side and 35 for the hypotenuse to solve for θ, the measure of the unknown angle:

sin^-1 (25 / 35) = θ

sin^-1 (5/7) = θ

45.5846914 = θ

46 = θ

Thus, the unknown angle is about 46°.

(8): When the unknown (?) angle is the reference angle:

• the side that is 23 units long is the adjacent side,
• and the side that is 28 units long is the hypotenuse.

Thus, we can find the measure of the unknown (?) angle in ° using the inverse cosine ratio which is given by:

cos^-1 (adjacent / hypotenuse) = θ, where

• θ reference angle

Thus, we plug in 23 for the adjacent side and 28 for the hypotenuse to solve for θ, the measure of the unknown angle:

cos^-1 (23 / 28) = θ

34.77194403 = θ

35 = θ

Thus, the measure of the unknown angle is about 35°.