Question

Найдите cos a и tg a, если sin a=1/3

Answer 1

1)sin^{2} a+cos^{2} a=1(основное тригонометрическое тождество)
(1\3)^{2} +cos^{2} a=1
cos^{2} a=1-1\9
cos^{2} a=8\9
cos a=√8\3
2)tg a=sin a\cos a
tg a=1*3\3*√8
tg a=1\√8

Answer 2

To find cos(a) and tan(a), we can use the Pythagorean identity and the definition of tangent in terms of sine and cosine.

Given that sin(a) = 1/3, we can find cos(a) using the Pythagorean identity:

cos^2(a) + sin^2(a) = 1

Substituting sin(a) = 1/3:

cos^2(a) + (1/3)^2 = 1 cos^2(a) + 1/9 = 1 cos^2(a) = 1 - 1/9 cos^2(a) = 8/9

Taking the square root of both sides:

cos(a) = ±√(8/9)

Since cos(a) is positive in the first and fourth quadrants, we take the positive square root:

cos(a) = √(8/9) = 2√2/3

To find tan(a), we can use the definition:

tan(a) = sin(a) / cos(a)

Substituting sin(a) = 1/3 and cos(a) = 2√2/3:

tan(a) = (1/3) / (2√2/3) tan(a) = (1/3) * (3/2√2) tan(a) = 1 / (2√2) tan(a) = √2 / (2 * 2) tan(a) = √2 / 4

Therefore, cos(a) = 2√2/3 and tan(a) = √2/4.