Question

Solve the right triangle. Side AB is 2.9, but I don't know how to find two missing angles with three sides and a 90-degree angle. Please explain your answer.

Answer 1

Answer: Choice (a) in the upper left corner

Angle A = 68 degrees

Angle B = 22 degrees

Side AB = 2.9 cm

The three values are approximate

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Explanation:

a = BC = 2.7 and b = AC = 1.1 are the two legs of the right triangle.

Use the pythagorean theorem to find the length of the hypotenuse

a^2 + b^2 = c^2

2.7^2 + 1.1^2 = c^2

8.5 = c^2

c^2 = 8.5

c = sqrt(8.5)

c = 2.91547594742266

c = 2.9

The hypotenuse is c = AB = 2.9

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We then use trig ratios to find the two missing angles.

Let's say we ignore the hypotenuse and focus on the two legs of the triangle.

With respect to reference angle A, side BC is opposite and AC is adjacent

Use the tangent ratio to get

tan(angle) = opposite/adjacent

tan(A) = BC/AC

tan(A) = 2.7/1.1

tan(A) = 2.45454545454546

A = arctan(2.45454545454546)

A = 67.8336541779176

A = 68

Note: arctan is the same as inverse tangent or $\tan^{-1}$. Make sure your calculator is in degree mode.

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Similarly,

tan(angle) = opposite/adjacent

tan(B) = AC/BC

tan(B) = 1.1/2.7

tan(B) = 0.4074074074074

B = arctan(0.4074074074074)

B = 22.166345822082

B = 22

Or you could use the value of A to get B = 90-A = 90-68 = 22. This works because A+B = 90 as we're working with a right triangle. In other words, A and B are complementary angles.

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In summary we found

A = 68

B = 22

AB = 2.9