find cosx and sinx if tanx = 4
sin ( x ) = + OR / [ tan (x) /sqrt ( 1 + tan ^2 (x)]
sin ( x ) = + OR - [ 4/ sqrt ( 1 + 4 ^2 ) ]
sin ( x ) = + OR - m4/ sqrt ( 1 + 16 )
sin ( x ) = + OR - 4/ sqrt ( 17)
cos ( x ) = + OR / 1 /sqrt ( 1 + tan ^2 (x)
cos( x ) = + OR - 1 / sqrt ( 17)
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cos ( x ) = + OR - 1 /sqrt ( 1 + tan ^2 (x) )
To find the values of cos(x) and sin(x) given that tan(x) = 4, we can use the trigonometric identity relating sin(x), cos(x), and tan(x).
The identity is: tan(x) = sin(x) / cos(x)
Since we know that tan(x) = 4, we can substitute this value into the identity:
4 = sin(x) / cos(x)
To solve for cos(x), we can rearrange the equation:
4 * cos(x) = sin(x)
cos(x) = sin(x) / 4
Now, we can use another trigonometric identity to find sin(x) in terms of cos(x):
sin^2(x) + cos^2(x) = 1
Substituting cos(x) = sin(x) / 4 into the identity:
sin^2(x) + (sin(x) / 4)^2 = 1
Multiplying through by 16 to eliminate the fraction:
16 * sin^2(x) + sin^2(x) = 16
Simplifying the equation:
17 * sin^2(x) = 16
sin^2(x) = 16 / 17
Taking the square root of both sides to solve for sin(x):
sin(x) = ±√(16 / 17)
Note that there are two possible values for sin(x) because sine is positive in the first and second quadrants.
To find cos(x), we can substitute the value of sin(x) back into the equation we found earlier:
cos(x) = sin(x) / 4
cos(x) = (±√(16 / 17)) / 4
cos(x) = ±√(16 / 17) / 4
So, the values of cos(x) are ±√(16 / 17) / 4, and the values of sin(x) are ±√(16 / 17).