# If the hypotenuse of a right triangle is 10 units and angle B is 70 degrees determine angle A and the other sides of the triangle.

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Given the triangle ABC such that:

B = 70 degrees.

Then The other angles are:

A = 90 degrees.

C = 20 degrees.

Then the hypotenuse is BC = 10 units.

Now we will calculate the length of the legs.

==> AC = BC*cos B = 10*cos70 = 10*0.3420 = 3.42 units

==> AB = BC*cos C = 10*cos20 = 10*0.9397 = 9.367 units.

Then the angles of the triangle are:

A = 90 degrees

B = 70 degrees.

C = 20 degrees.

The length of the sides are:

AB = 9.367 units.

BC = 10 units.

AC = 3.42 units.

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We have a right angled triangle and one of the sides given as 70 degrees. Now the second side is 180 - 90 - 70 = 20 degrees.

Also, the hypotenuse is 10. Let the other sides be a and b.

So cos 70 = a / 10 =.3420

a = .3420*10 = 3.420

cos 20 = b/ 10 = .9396

b = 0.9396*10 = 9.396

Therefore the sides of the triangle are 10, 3.42 and 9.396. The angles of the triangle are 90, 70 and 20 degrees.

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Let's suppose that the right angle is A = 90 degrees. The hypotenuse is opposite side to the right angle.

That means that B+C = 70 + C = 90

C = 90 - 70

C = 20 and B = 70

To determine the lengths of the other cathetus, we'll apply the Pythagorean theorem and the sine function.

The sine function is a ratio between the opposite cathetus and the hypothenuse. We'll note the cathetus as x and y.

sin B = x/10

sin 70 = x/10

x = 10*sin 70

x = 10*0.93

x = 9.4 units

We'll apply Pythagorean theorem in a right angle triangle:

10^2 = x^2 + y^2

100 = 88.36 + y^2

y^2 = 100 - 88.36

y^2 = 11.64

y = 3.4 units

We'll keep only the positive value for y, since it is the length of the cathetus of the right angle triangle and it cannot be negative.