Answer:

Step-by-step explanation:

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Given:

Triangle ABC, where angle B = 90 degrees

AB = 40

AC + BC = 80

To find:

sin A

cos A

tan C

Solution:

Sin A

Since angle B is 90 degrees, triangle ABC is a right triangle. The opposite side to angle A is AB = 40, and the hypotenuse is AC + BC = 80.

sin A = opposite/hypotenuse = AB/(AC + BC) = 40/80 = 0.5

Cos A

The adjacent side to angle A is BC. We don't know the value of BC, so we can't find cos A.

Tan C

C is the angle opposite to hypotenuse AC + BC. We know the values of AC + BC and AB, so we can find tan C.

tan C = opposite/adjacent = AB/BC = 40/(80 - 40) = 40/40 = 1

Therefore, sin A = 0.5, cos A is unknown, and tan C = 1.

Given that in triangle ABC, angle B is 90 degrees and AB = AC + BC, we can solve for the other values:

(i) To find sin A:

In a right triangle, sin A = opposite / hypotenuse.

Here, AC is the opposite side to angle A and AB is the hypotenuse.

So, sin A = AC / AB.

(ii) To find cos A:

In a right triangle, cos A = adjacent / hypotenuse.

Here, BC is the adjacent side to angle A and AB is the hypotenuse.

So, cos A = BC / AB.

(iii) To find tan C:

In a right triangle, tan C = opposite / adjacent.

Here, AC is the opposite side to angle C and BC is the adjacent side.

So, tan C = AC / BC.

Given the relationship between AC and BC, you can substitute the value of AC from the given equation (AC + BC = 80) to solve for BC, and then use that to find the trigonometric values as described above.