Find the unknown side length in the given triangle. the two angles of the right triangle are 20 and 30

Question

Round to the nearest hundredth.

Answer 1

Since the given triangle is a right triangle, the sum of the two angles must be 90 degrees. Therefore, the third angle can be found by subtracting the sum of the other two angles from 90 degrees: 90 - 20 - 30 = 40 degrees.

Let's label the unknown side length as x. We can use the sine function to solve for x:

sin(20) = opp/hyp (opposite side of angle 20 / hypotenuse)
sin(20) = x/hyp (since the opposite side of angle 20 is x)
hyp = x/sin(20) (cross-multiplying)

Similarly, we can use the cosine function to solve for the same unknown side length:

cos(30) = adj/hyp (adjacent side of angle 30 / hypotenuse)
cos(30) = x/hyp (since the adjacent side of angle 30 is also x)
hyp = x/cos(30)

Setting these two expressions equal to each other:

x/sin(20) = x/cos(30)

Now, we can solve for x by cross-multiplying:

cos(30) * x = sin(20) * x

Dividing both sides by x:

cos(30) = sin(20)

Now, we can solve for x by dividing both sides by cos(30):

x = (sin(20) / cos(30))

Using a calculator to evaluate this expression to the nearest hundredth:

x ≈ 0.3726

Therefore, the unknown side length is approximately 0.37.