# Sin To Cos Formula

## Sin To Cos Formula

The Sin To Cos Formula, which relates to the angles and side ratios of a right-angled triangle, are the fundamental trigonometric functions. The ratio of the adjacent side to the hypotenuse is known as the cosine of an angle, and the sine of an angle is the ratio of the opposite side to the hypotenuse. These establish the fundamental identifies for sharp angles. The trigonometric function is the extension of these ratios to any angle in terms of radian measure. In the first and second quadrants, sin is positive, whereas the first and fourth quadrants of cos are positive. Under the real number domain, the Sin To Cos Formula has a range of [-1,1].

The Pythagorean theorem states that x2 + y2 = 1, where x and y are the lengths of the right-angled triangle’s legs, holds true if (x,y) is a point on the unit circle and a ray from the origin (0, 0) to (x, y) forms an angle of from the positive axis. As a result, cos2 + sin2 = 1 replaces the original Sin To Cos Formula.

The Sin To Cos Formula identities that are used in trigonometric functions have various applications. The Sin To Cos Formula makes it easier to assess all trigonometric expressions. The Extramarks website and mobile application are good places for students to search for the in-depth discussion.

## Examples of Sin to Cos formula

Example 1: When, sin X = 1/2 and cos Y = 3/4 then find cos(X+Y)

Solution: We know cos(X + Y) = cos X cos Y – sin X sin Y

Given sin X = 1/2

We know that, cos X = √(1 – sin2X) = √(1 – (1/4)) = √3/2

Thus, cos X = √3/2

Given cos Y = 3/4

We know that, sin Y = √(1 – cos2Y) = √(1 – (9/16)) = √7/4

Thus, sin Y = √7/4

cos X = √3/2, and sinY = √7/4

Applying the sum of cos formula, we have cos(X+Y) = (√3/2) × (3/4) –  1/2 × (√7/4)

= (3√3 – √7)/8

Example 2: If sin θ = 3/5, find sin2θ.

Solution: We know that sin2θ = 2 sin θ cos θ

We need to determine cos θ.

Students can use the sin cos formula cos2θ + sin2θ = 1.

Rewriting, we get cos2θ = 1 –  sin2θ

= 1-(9/25)

cos2θ = 16/25

cos θ = 4/5

sin2θ = 2 sin θ cos θ

= 2 × (3/5) × (4/5) = 24/25