The diagonal of a square is 5√2. What is the sine and cosine of the angle that it makes with its side?

Asked on by gretjxfj

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jerichorayel's profile pic

jerichorayel | College Teacher | (Level 2) Senior Educator

Posted on

first we try to get the value of the sides.

we take the diagonal to form a right triangle.

c^2 = a^2 + b^2        since a = b ( for square)

c^2 = 2a^2

(5√2)^2 = 2a^2 

a = 5 and b = 5

since it is a square, we know that the angles of the sides are 90.



is it alright for you to check the question.

probably theres more into it.


but remember the sohcahtoa

sin a = 5 /(5√2 )

a = 45 degrees

cos a = 5 /(5√2 )

a = 45 degrees

sciencesolve's profile pic

sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted on

You should remember that the diagonals of a square are bisectors of the angles, hence, the right triangles the diagonals form are isosceles.

Since the diagonals are the bisectors of angles, the bisected angles measure 45 degrees.

You should remember that the sine and cosine of `45^o `  have equal values.

You also should remember the definitions of sine and cosine functions such that:

`sin 45^o`  = opposite leg/hypotenuse

`cos 45^o ` = adjacent leg/hypotenuse

Since the legs of triangle are also the sides of square, you should come up with the following notation for the side of square, such that:


`sin 45^o`  = x/hypotenuse

`cos 45^o`  = x/hypotenuse

The problem provides the length of diagonal of the square that is the hypotenuse of right triangles, also.

hypotenuse = `5sqrt2`

You need to remember that `sin 45^o = cos 45^o = sqrt2/2`  such that:

`sqrt2/2 = x/(5sqrt2) =gt x = 5`

Hence, evaluating the sine and cosine of the angle that diagonal of square makes to its sides, yields `sin 45^o = cos 45^o = sqrt2/2.`


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