Here we will find the relation between sin θ and cos θ. Every trigonometric ratio have a relation to the other trigonometric ratio. It may be more than one relation between two trigonometric ratio. It depends on we that how we take both trigonometric ratio and solve it.
There is also a theorem exist. the theorem says that, The sin of any acute angle is equal to the cosine of its complement. this theorem is also proved in the below of this article.
Today, In this post we will solve and make a relation between sin and cos. If anyone has doubt to understand the relation between sin θ and cos θ. you can watch the below youtube video for better understanding.
relation between sin θ and cos θ
The relation between sin θ and cos θ is that In a right angled triangle the sin of an acute angle is equal to the cos of another acute angle. on the other hand cos of an acute angle is equal to the sin of its complement.
please note that the sum of the angles of sin and cos must be complement angle. that means if we add the angle of sin and angle of cos then the result is 90 degree.
lets take an example to understand the relation between sin and cos.
In the given figure, there is a right angled triangle ABC. angle A and c are acute angle and angle B is a right angle called 90°.
lets take angle angle C firstly.
sin 30° = p/h = AB/AC————-(1)
Now, Lets take angle A.
cos 60° = b/h = AB/AC ————(2)
Now, from equation 1 and 2 we get,
sin 30° = cos 60°
we can write cos 60° as cos (90° – 30°). so ,
sin 30° = cos (90° – 30°), or
sin θ = cos (90° – θ)
Hence, the mathematical relation between sin and cos is sin θ = cos (90° – θ).
the theorem also proved that, The sin of any acute angle is equal to the cosine of its complement.
sin θ = cos (90° – θ).
similarly,
cos θ = sin (90° – θ)
Example to show the relation between sin θ and cos θ.
Q.If sin(3x + 10)º = cos(x + 24)º, find x.
solution:-
sin(3x + 10)º = cos(x + 24)º
we know that sin of any acute angle is equal to the cosine of its complement.
and also the sum of the both angle of the sin and angle of the cosine is equal to 90 degree.
therefore,
(3x + 10)º + (x + 24)º = 90°
3x + x + 24° + 10° = 90°
4x + 34° = 90°
4x = 90° – 34° = 56°
x = 56/4 = 14°
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