Cos 45° = Cos π/4 = 1/√2

In trigonometry, the three main ratios room sine, cosine and tangent. If the trigonometric proportion of any kind of angle is taken because that a best angled triangle, climate the values rely on sides of the triangle. Cos of edge is same to the ratio of the adjacent side and hypotenuse. Cos θ = adjacent Side/Hypotenuse

## Cos 45° Value

The specific value the cos 45 degrees is 1/√2 (in surd form), i m sorry is also equal come sin 45 degrees. The is one irrational number, equal to 0.7071067812… in decimal form. The approximate value of cos 45 is same to 0.7071.

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 Cos 45° = 1/√2 = √2/2

Therefore, 0.7071 or 1/√2 is a value of a trigonometric duty or trigonometric ratio of conventional angle (45 degrees).

## Proof

Suppose we have actually a right-angled triangle, in i m sorry the various other two angles room equal to 45 degrees. Now, if the angles of a best triangle space 45 degrees, climate the nearby sides are equal in length. Let us take the size of surrounding sides equal to ‘l’ and hypotenuse is ‘r’.

According to the Pythagoras theorem, we recognize that, Hypotenuse2= Perpendicular2 + nearby Side2

PQ2 = PR2 + QR2

⇒ r2 = l2 + l2

⇒ r = √2 l

⇒ l/r = 1/√2

Thus, length of adjacent side/Hypotenuse = 1/√2

Therefore, we have the right to say, Cos 45° = 1/√2

Hence, proved.

## Cos 45° and Sin 45°

We can also prove the worth of cos 45° v a trigonometry approach.

As we know, sin 45° = 1/√2

Also, by trigonometric identities, we recognize that,

Sin2 x + cos2 x = 1

Or Cos2 x = 1 – sin2 x

Put x = 45°

Cos2 45° = 1 – sin2 45°

Put the worth of sin 45° in the over equation.

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Cos2 45° = 1 – (1/√2)2

Cos2 45° = 1 – ½

Cos 45° = √1/2 = 1/√2

Hence, we acquired the worth of cos 45°.

## Cos proportion Table

 Cos 0° 1 Cos 30° or Cos π/6 2 Cos 45° or Cos π/4 1/√2 Cos 60°or Cos π/3 √3/2 Cos 90° or Cos π/2 0 Cos 120° or Cos 2π/3 -1/2 Cos 150° or Cos 5π/6 √3/2 Cos 180° or Cos π -1 Cos 270° or Cos 3π/2 0 Cos 360°or Cos 2π 1

## Solved Examples

Question 1: discover the worth of cos 45° + sin 30°

Solution: Given, cos 45° + sin 30° cos 45° + sin 30° = 1/√2 + ½ = (√2+1)/2

Question 2: Evaluate: 2 sin 60° – 4 cos 45°

Solution: Given, 2 sin 60° – 4 cos 45°

Since, sin 60° = √3/2 and also cos 45° = 1/√2

Therefore, ⇒ 2 (√3/2) – 4 (1/√2)

⇒ √3 – 2√2

Question 3: discover Cos 45 + Cos 90.

Solution: Cos 45° = 1/√2 Cos 90° = 0

Therefore, Cos 45° + cos 90° = 1/√2

## Practice Questions

Evaluate 2 cos 45° + cos 0°Find the worth of sin 60° – cos 45°.Tan 45° – cos 45° = ?Cos 45° – ½ (Cot 45°) = ?