Question

Найти значения: cos a, ctg a, если: sina = 5/13 и п/2 <а <п​

Answer 1

$1) cosa=\sqrt{1-(\frac{5}{13})^{2} } =-\frac{12}{13} \\2) ctga=\frac{\frac{12}{13} }{\frac{5}{13} } =- \frac{12}{5}$

ставим отрицательные знаки, т.к. значения косинуса и котангенса во 2 четверти отрицательны

Answer 2

To find the values of cos a and ctg a, given that sin a = 5/13 and π/2 < a < π, we can use trigonometric identities to relate different trigonometric functions.

Given that sin a = 5/13, we can find cos a using the Pythagorean trigonometric identity:

sin^2 a + cos^2 a = 1

Substitute sin a = 5/13:

(5/13)^2 + cos^2 a = 1

25/169 + cos^2 a = 1

Now, solve for cos a:

cos^2 a = 1 - 25/169

cos^2 a = (169 - 25)/169

cos^2 a = 144/169

cos a = ±√(144/169) (Taking the square root of both sides, and since a is in the second quadrant, cos a is negative)

cos a = -12/13

Next, to find ctg a (cotangent of a), we can use the reciprocal trigonometric identity:

cot a = 1/tan a

Since we know sin a = 5/13, and tan a = sin a / cos a, we can find ctg a:

tan a = sin a / cos a

tan a = (5/13) / (-12/13)

tan a = -5/12

Now, find ctg a:

cot a = 1 / tan a

cot a = 1 / (-5/12)

cot a = -12/5

So, the values of cos a and ctg a are:

cos a = -12/13 ctg a = -12/5