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.

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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/62
Cos 45° or Cos π/41/√2
Cos 60°or Cos π/3√3/2
Cos 90° or Cos π/20
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/20
Cos 360°or Cos 2π1

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°) = ?