# Sin Cos Formulas

## Sin Cos Formulas

The Sin Cos Formulas, which relate 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 Cos Formulas have a range of [-1,1].

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## What Are Sin Cos Formulas?

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 Cos Formulas. In trigonometry, the Sin Cos Formulas are used to calculate angles and side ratios of right-angled triangles.

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The Sin Cos Formulas that are used in trigonometric functions have various applications. These trigonometric formulae make it easier to assess all trigonometric expressions. Students can go through them in more depth on the Extramarks website and mobile application.

### Sin Cos Formulas

The roles of negative angles are: for any sharp angle of;

• sin(-) equals – sin
• cos (-) equals cos

Trigonometry function identities expressed in terms of their complements:

• cos = sin(-90°)
• cos(90° -) = sin

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### Sum and Difference of Sin Cos Formulas

Compound angles are those formed by adding or subtracting two or more other angles. The compound angles are written by students as and. There are Sin Cos Formulas for extending or condensing trigonometric expressions with regard to compound angles. Students can go through the same on the Extramarks website and mobile application.

• sin (α + β) = sin α cos β + cos α sin β
• sin (α – β) = sin α cos β – cos α sin β
• cos (α + β) = cos α cos β – sin α sin β
• cos (α – β) = cos α cos β + sin α sin β

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Students from other boards can also refer to the Sin Cos Formulas notes and solutions.

## Transformation of Sin Cos Formulas

students should choose a few identities from one side to use, and then substitute others until the side is changed into the opposite side. Rewrite either side of the equation and translate it to the other side in order to confirm a person’s identification. Get the product-to-sum and sum-to-product formulae from the aforementioned sum and difference identities.

When given a product of cosines, product-to-sum formulae are used. Students must write the formula, replace the provided angles, and then simplify the product as a sum or difference.

• 2 sin α cos β = sin (α +β) + sin (α – β)
• 2 cos α sin β = sin (α + β) – sin (α – β)
• 2 cos α cos β = cos (α + β) + cos (α – β)
• 2 sin α sin β = cos (α – β) – cos (α + β)

Sums of sine or cosine can be expressed as products using the sum-to-product formulas. The formulas for these are as follows:

• sin α + sin β = 2 sin((α+β)/2) cos((α−β)/2)
• sin α – sin β = 2 cos((α+β)/2) sin((α−β)/2)
• cos α + cos β= 2 cos((α+β)/2) cos((α−β)/2)
• cos α – cos β = -2 sin((α+β)/2) sin((α-β)/2)

Designed by Extramarks experts, Sin Cos Formulas notes and solutions provide a very simple yet powerful framework that is extremely easy to understand and student-friendly.

### Derivation of Product to Sum Formulas

Here, the sum of the cosine and sine products are used. The sum and difference identities for cosine allow us to construct the product-to-sum formula. The result of combining the two equations is cos cos + sin sin = cos( ) + cos cos sin sin = cosα cosβ + sinα sinβ = cos(α − β)

+ cosα cosβ − sinα sinβ = cos(α + β)

––—————————-––————-

2

cosα cosβ = cos(α−β) + cos(α + β)

Divide the result by 2, then find the product of cosines: cos cos equals (1/2) [cos(α−β) + cos(α+β)]

Similarly, by expressing the products as the sum/difference, students may get the other formulae.

### Derivation of Sum to Product Formulas

There are several issues that call for the product’s reverse to sum. Let’s examine how these sum-to-product formulae were developed. Students should utilize the replacements (u+v)/2 = and (u-v)/2 = for this.

After that, u – u = [(u + v)/2] u – u = [(u + v)/2] – [(u- v)/2] = v

They should then calculate the formula for the sum to the product. In the product-to-sum formula, swap out and think about

(sinα cosβ) = (1/2)[sin(α + β) + sin(α – β)]

Substituting for (α + β) and αβ, we get

sin((u+v)/2) cos ((u-v)/2) = 1/2[sinu + sin v]

2sin((u+v)/2)) cos ((u-v)/2) = sinu + sin v

The additional identities for sum-to-product may be derived in a similar manner.

### Sin Cos Formulas of Multiple Angles

The double and triple angles formula and the half-angle formulas as follows:

• sin 2θ = 2 sinθ cosθ
• sin 3θ = 3 sinθ – 4 sin3θ
• cos 2θ = cos2 θ – sin2 θ
• cos 2θ = 2cos2θ – 1
• cos 2θ = 1- 2sin2 θ
• cos 3θ = 4 cos3θ – 3cosθ
• sin (θ/2) = ± √((1- cosθ)/2)
• cos (θ/2) = ± √((1+ cosθ)/2)
• sin θ = 2tan (θ/2) /(1 + tan2 (θ/2))
• cos θ = (1-tan2 (θ/2))/(1 + tan2 (θ/2))

## Examples Using Sin Cos Formulas

Example 1:

If sin X = 1/2 and cos Y = 3/4 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

Answer: cos(X+Y) = (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.

Let us 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