If 13cos A=12,find the value of sec A + sin A
cos A = 12/13
sec A = 1/cos A = 13/12
sin A =sqrt ( 1 - cos^2 A) = sqrt (169/169 - 144/169) = sqrt( 25/169) =5/13
13/12 + 5/13 = (169 +60) / 156 = 229/156
or ...
13cos A=12
cosA = 12/13, recognize the right-angled triangle with sides 5-12-13
so sinA = 5/13
since cosA is positive,
A could be in quad I or IV
if A is in I, cosA = 12/13, sinA = 5/13,
then sec A + sin A = 13/12 + 5/13 = 228/156
If A is in IV, cosA = 12/13, sinA = -5/13, then
secA + sinA = 13/12 - 5/13 = 109/156
To find the value of sec A + sin A, we need to determine the values of sec A and sin A first.
Given: 13cos A = 12
To find the value of cos A, we can rewrite the equation as:
cos A = 12/13
Now, recall that sec A is the reciprocal of cos A:
sec A = 1/cos A
Substituting the value of cos A we found earlier:
sec A = 1/(12/13)
sec A = 13/12
Next, we need to find the value of sin A using the Pythagorean identity:
sin^2 A + cos^2 A = 1
Since we know the value of cos A, we can substitute it into the equation:
sin^2 A + (12/13)^2 = 1
sin^2 A + 144/169 = 1
sin^2 A = 1 - 144/169
sin^2 A = (169 - 144)/169
sin^2 A = 25/169
Taking the square root of both sides:
sin A = √(25/169)
sin A = 5/13
Finally, we can find sec A + sin A by adding the values we found:
sec A + sin A = 13/12 + 5/13
To simplify this expression, we need to find a common denominator:
sec A + sin A = (13/12)*(13/13) + (5/13)*(12/12)
sec A + sin A = 169/156 + 60/156
sec A + sin A = (169 + 60)/156
sec A + sin A = 229/156
Therefore, the value of sec A + sin A is 229/156.
To find the value of sec A + sin A, we need to find the values of sec A and sin A separately, and then add them.
Given that 13cos A = 12, we can start by finding the value of cos A. Divide both sides of the equation by 13:
cos A = 12/13
Now, to find the value of sec A, we need to use the reciprocal of cosine, which is secant. So, sec A = 1/cos A. Substituting the value of cos A that we just found:
sec A = 1/(12/13)
= 13/12
Next, let's find sin A. Recall the Pythagorean identity: sin^2 A + cos^2 A = 1. Since we have the value of cos A, we can solve for sin A:
sin^2 A + (12/13)^2 = 1
sin^2 A + 144/169 = 1
sin^2 A = 1 - 144/169
sin^2 A = 25/169
Taking the square root of both sides:
sin A = ±(√25/√169)
sin A = ±5/13
Now we have the values of sec A and sin A. To find the value of sec A + sin A, we substitute the respective values:
sec A + sin A = (13/12) + (5/13)
To simplify, we need a common denominator:
sec A + sin A = (13/12) + (5/13) = (169/156) + (60/156) = 229/156
Therefore, the value of sec A + sin A is 229/156.