Why is cos 45˚ equal to √2/2? Why doesn't is cos 45˚ equal 1/√2?
2007-01-09 06:19:06 · 6 answers · asked by thomasgraham880 1 in Science & Mathematics ➔ Mathematics
They're the same: sqrt(2) / 2 is identically equal to 1 / sqrt(2), or 0.70710678... .
How you write them is simply a matter of choice. Some people are uncomfortable when square roots are in the denominator; shifting them "upstairs" into the numerator reduces their discomfort.
Geometrically, you could have (45, 45, 90) deg. triangles with sides (sqrt(2), sqrt(2), 2), OR (1, 1, sqrt(2)). (Correspondingly ordered sides are OPPOSITE the ordered angles.) These triangles have the same shape and are simply size scalings of one onother by [sqrt(2)]^(+/-1).
But basically, whether you look at these "two" values from an arithmetical or a geometrical/trigonometrical point of view, they are absolutely identical.
Live long and prosper.
2007-01-09 06:22:13 · answer #1 · answered by Dr Spock 6 · 1⤊ 0⤋
For 45 degrees, the triangle is isoceles. The two legs of the right triangle have the same length, call it x. For trig funcitions, the hypotenuese is 1.
From pathagarians theorom,
x^2 + x^2 = 1
2x^2 = 1
x^2 = 1/2
x = 1/sqrt(2)
Multiply top and bottom by sqrt(2) and
x = sqrt(2)/2
Bozo
2007-01-09 06:27:12 · answer #2 · answered by bozo 4 · 0⤊ 0⤋
1/â2 can be rationalized. Math people don't like to use radicals in denominators. So, multiply 1/â2 by â2/â2 to give â2/â4 = â2/2.
2007-01-09 06:24:08 · answer #3 · answered by Nicknamr 3 · 1⤊ 0⤋
â2/2=1/â2
2=(â2)^2 by definition of â
(â2)/(â2)*â2) cancel â2 =1/â2
2007-01-09 06:24:54 · answer #4 · answered by yupchagee 7 · 1⤊ 0⤋
1 / sqrt(2) = sqrt(2) / 2
they are equal
2007-01-09 06:26:57 · answer #5 · answered by Rick 5 · 1⤊ 0⤋
â2/2=.707
1/â2=.707
They are both the same answer.
2007-01-09 08:01:40 · answer #6 · answered by Anonymous · 0⤊ 0⤋