If sin x = .21,what is cos x?
I need help with my homework! If sin(x) = 0.21, what is cos(x)? Here are the answer choices:
a) 0.76
b) 0.98
c) 0.87
d) 0.65
6 Answers
Start with the following trigonometric identity: sin^2(x) + cos^2(x) = 1
sin^2(x) + cos^2(x) = 1
cos^2(x) = 1 – sin^2(x)
cos(x) = (+/-)sqrt(1 – sin^2(x))
cos(x) = (+/-)sqrt(1 – .21^2) = +/-0.98, which matches choice b. However, this is only the case if x is in Quadrant I, i.e., 0 to pi/2 radians or 0 to 90 degrees. You could have sin(x) = .21 with x in Quadrant II, i.e., pi/2 to pi radiants or 90 to 180 degrees, and then cos(x) would be -.98, not .98.
Sin x = 0.21 = 21/100 = o/h
To find the adjacent ‘a’ use Pythagoras.
21^2 + a^2 = 100^2
a^2 = 100^2 – 21^2
a^2 = (100 – 21)(100 + 21)
a^2 = 79 x 121
a^2 = 9559
a = sqrt (9559) = 97.77…
Cos x = a/h = 97.77.. / 100
Cos x = 0.977777…
Cos x = 0.98 ==> ‘b’
Since sin x = .21, the opposite side equals .21 and the hypotenuse is 1.
By the Pythagorean Theorem, the adjacent side is â(1^2 – .21^2) = 0.978
So, cos x = 0.978 / 1 = 0.978
Sine squared plus cosine squared = 1, so:
0.21^2 = 0.0441
1 – 0.0441 = 0.9559
0.9559^(1/2) = 0.9777
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