Given the right triangle ABC, where A=90 degrees, calculate the expression: E=cosB/sinC+ cosC/sinB
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We have a right triangle with A = 90 degrees and we have to calculate cos B / sin C + cos C / sin B
Let's take the sides opposite angle A as a, that opposite angle B as b and that opposite angle C as c.
cos B / sin C + cos C / sin B
=> [(c / a)/(c/ a)] + [(b/a)/(b/a)]
=> 1 + 1
=> 2
The expression = 2.
We'll calculate the first ratio: cosB/sinC
We know that A = pi/2 and the sum of the angles of a triangle is pi.
A + B + C = PI
pi/2 + B + C = pi
B + C = pi - pi/2
B +C = pi/2
B = pi/2 - C
Now, we'll apply cosine function both sideS:
cos B = cos (pi/2 - C)
cos B = cos pi/2*cos C + sin pi/2*sin C
cos pi/2 = 0 and sin pi/2 =1
cos B = sin C
cosB/sinC = sin C/sin C = 1
Now, we'll calculate the second ratio:
cosC/sinB = cos (pi/2 - B)/sin B
cosC/sinB = sin B/sin B
cosC/sinB = 1
The value of the given expression is:
E = 1 + 1
E = 2
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