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Use the right triangle and the given information to solve the triangle.

Given:
a = 8
B = 62 degrees

Find:
b, c, and A.

Answer :

The values are: a = 8 B = 62°

We can find c using the Pythagorean theorem, which states that for a right triangle with legs a and b and hypotenuse c, a² + b² = c². We can say that since a and c are the legs of the right triangle, and b is the hypotenuse. Using the Pythagorean theorem, we have:

b² = a² + c²

We are given the value of a to be 8, so we can substitute this value into the above equation:

b² = 8² + c²b² = 64 + c²We are looking for the values of b, c, and A.

We know the value of B to be 62°, so we can use the fact that the sum of the angles in a triangle is 180° to find the value of A. We have:

A + B + C = 180°

A + 62° + 90° = 180°

A + 152° = 180°

A = 180° - 152°

A = 28°

Therefore, we have:

A = 28°B

= 62°c²

= b² - a²c²

= b² - 64A

= 28°b²

= c² + a²b²

= c² + 64

Since we have two equations for b², we can equate them:

c² + 64 = b²Substitute c² in terms of b² obtained from the Pythagorean theorem:

c² = b² - 64c² + 64

= b²

Substitute

A = 28°, B = 62°, and C = 90° in the trigonometric ratio to obtain the value of b:

b/sin B = c/sin C

b/sin 62° = c/sin 90°b = c sin 62°

Substitute c² + 64 = b² into the above equation:

c sin 62°

= √(c² + 64) sin 62°c

= √(c² + 64) tan 62°

Square both sides to obtain:

c² = (c² + 64) tan² 62°c²

= c² tan² 62° + 64 tan² 62°

c² - c² tan² 62°

= 64 tan² 62°

Factor out c²:c²(1 - tan² 62°) = 64 tan² 62°

Divide both sides by (1 - tan² 62°):

c² = 64 tan² 62° / (1 - tan² 62°)c²

= 137.17c ≈ √137.17c ≈ 11.71

Substitute this value of c into the equation obtained for b:

b = c sin 62°

b = 11.71

sin 62°b

≈ 10.37.

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