Look at this geometry figure:
In all these right angled triangles of different sizes, <A has the same measure. Since <B = 90o, we have <C = 90o - <A, in each of these triangles. So, all these triangles have equal angles and hence are similar. Thus even though the lengths of the sides BC, CA, AB are different in these, the numbers `(BC)/(AC)` and `(AB)/(AC)` are the same for all these triangles.
In all these triangles, AC is the hypotenuse. The side BC is the opposite side of <A. The third side AB is called the adjacent side of<A. In a right angled triangle with <A as one of its angles, the number obtained by dividing the adjacent side of <A by the hypotenuse is called the cosine of <A and its written cos A. Thus
cos A = `(adjacent- side- of- angleA) / ( hypotenuse )`
Note that cos A is a number indicating the size of <A alone and not depend on the size of the right angled triangle containing <A.
Examples for Triangle Geometry Cosine:
Example 1:
Determine the cosine of angle A or cos A; If <A =45o in the right angled triangle ABC, as in the figure below
then, as seen earlier, we have
`(BC)/(AC)` = `(AB )/(AC)` =1/`sqrt(2)`
which in our new terminology can be written
cos 45o = 1/`sqrt(2)`
Example 2:
Determine the cos30o, if <A =30o in the right angled triangle ABC, as in the figure below
Solution:
By definition, we know that cos A = `(AB)/(AC)`
Therefore, cos 30 = `sqrt(3)` / 2
Example 3:
Measure the cos C, if <C =60o in the right angled triangle ABC, as in the figure below
Solution:
Given:
`Delta` ABC , AB = `sqrt(3)` , BC = 1 and AC = 2.
To find the triangle geometry cosine C:
By definition, we know that, cos C = `(BC)/(AC)`
That is cos 60o = `1/2`
This is the required triangle geometry cosine values.
Example 4:
Measure the cosine of angle A or cos A; If <A =20o, AB = 4.7 and AC =5 in the right angled triangle ABC, as in the figure below
Solution:
Given:
In `Delta` ABC, <A = 20o , AB = 4.7 and AC =5.
To find the triangle geometry cosine of angle A:
We have found by actual measurement that in such a triangle as `Delta` ABC
`(AB)/(AC)` = `(4.7) /(5)` = 0.94
This means in other words
cos 20o = 0.94
In all these right angled triangles of different sizes, <A has the same measure. Since <B = 90o, we have <C = 90o - <A, in each of these triangles. So, all these triangles have equal angles and hence are similar. Thus even though the lengths of the sides BC, CA, AB are different in these, the numbers `(BC)/(AC)` and `(AB)/(AC)` are the same for all these triangles.
In all these triangles, AC is the hypotenuse. The side BC is the opposite side of <A. The third side AB is called the adjacent side of<A. In a right angled triangle with <A as one of its angles, the number obtained by dividing the adjacent side of <A by the hypotenuse is called the cosine of <A and its written cos A. Thus
cos A = `(adjacent- side- of- angleA) / ( hypotenuse )`
Note that cos A is a number indicating the size of <A alone and not depend on the size of the right angled triangle containing <A.
Examples for Triangle Geometry Cosine:
Example 1:
Determine the cosine of angle A or cos A; If <A =45o in the right angled triangle ABC, as in the figure below
then, as seen earlier, we have
`(BC)/(AC)` = `(AB )/(AC)` =1/`sqrt(2)`
which in our new terminology can be written
cos 45o = 1/`sqrt(2)`
Example 2:
Determine the cos30o, if <A =30o in the right angled triangle ABC, as in the figure below
Solution:
By definition, we know that cos A = `(AB)/(AC)`
Therefore, cos 30 = `sqrt(3)` / 2
Example 3:
Measure the cos C, if <C =60o in the right angled triangle ABC, as in the figure below
Solution:
Given:
`Delta` ABC , AB = `sqrt(3)` , BC = 1 and AC = 2.
To find the triangle geometry cosine C:
By definition, we know that, cos C = `(BC)/(AC)`
That is cos 60o = `1/2`
This is the required triangle geometry cosine values.
Example 4:
Measure the cosine of angle A or cos A; If <A =20o, AB = 4.7 and AC =5 in the right angled triangle ABC, as in the figure below
Solution:
Given:
In `Delta` ABC, <A = 20o , AB = 4.7 and AC =5.
To find the triangle geometry cosine of angle A:
We have found by actual measurement that in such a triangle as `Delta` ABC
`(AB)/(AC)` = `(4.7) /(5)` = 0.94
This means in other words
cos 20o = 0.94
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