Cos 45 Cos 15

Cos 45 Cos 15.

The value of cos 45° is equal to 1/√ii.

In trigonometry, the three primary ratios are sine, cosine and tangent. If the trigonometric ratio of any angle is taken for a right angled triangle, so the values depend on sides of the triangle. Cos of bending is equal to the ratio of the next side and hypotenuse. Cos θ = Adjacent Side/Hypotenuse

Cos 45° Value

The verbal value of cos 45 degrees is i/√2 (in surd form), which is besides equal to sin 45 degrees. It is an irrational number, equal to 0.7071067812… in decimal course. The approximate value of cos 45 is equal to 0.7071.

Therefore, 0.7071 or i/√2 is a value of a trigonometric function or trigonometric ratio of standard angle (45 degrees).

Proof

Suppose we take a right-angled triangle, in which the other 2 angles are equal to 45 degrees. At present, if the angles of a right triangle are 45 degrees, and then the side by side sides are equal in length.

Let us take the length of adjacent sides equal to ‘l’ and hypotenuse is ‘r’.

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According to the Pythagoras theorem, we know that, Hypotenuse2= Perpendicular2
+ Next Sidetwo

PQ2
= PR2
+ QRii

⇒ r2
= l2
+ lii

⇒ r = √2 l

⇒ l/r = 1/√2

Thus, Length of adjacent side/Hypotenuse = i/√2

Therefore, we tin can say, Cos 45° = 1/√2

Hence, proved.

Cos 45° and Sin 45°

We can also prove the value of cos 45° with a trigonometry approach.

As we know, sin 45° = 1/√ii

Likewise, by trigonometric identities, nosotros know that,

Sinii
ten + cos2
x = 1

Or Cos2
x = ane – sinii
ten

Put 10 = 45°

Cos2
45° = 1 – sinii
45°

Put the value of sin 45° in the in a higher place equation.

Cos2
45° = 1 – (1/√2)2

Cos2
45° = 1 – ½

Cos 45° = √i/2 = ane/√2

Hence, we got the value of cos 45°.

Cos Ratio Table

Cos 0° i
Cos 30° or Cos π/6 2
Cos 45° or Cos π/iv i/√two
Cos 60°or Cos π/three √3/2
Cos xc° or Cos π/two
Cos 120° or Cos 2π/three -1/2
Cos 150° or Cos 5π/half dozen √3/ii
Cos 180° or Cos π -1
Cos 270° or Cos 3π/ii
Cos 360°or Cos 2π 1

Solved Examples

Question one: Find the value of cos 45° + sin 30°

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

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

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

Since, sin 60° = √three/2 and cos 45° = ane/√2

Therefore, ⇒ 2 (√3/ii) – four (1/√2)

⇒ √3 – 2√ii

Question 3: Find Cos 45 + Cos 90.

Solution: Cos 45° = i/√2 Cos xc° = 0

Therefore, Cos 45° + cos 90° = ane/√2

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Practice Questions

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

Frequently Asked Questions on Cos 45 Degrees

What is the exact value of cos 45 degrees?

The exact value of cos 45 degrees is ane/√2 or √2/ii.

Is cos 45 and sin 45 the same?

Yes, the value of cos 45 and sin 45 is the same.

What is the value of cos xc?

The value of cos xc is equal to 0.

What is the sum of cos 45 and sin 45?

Cos 45 + Sin 45 = 1/√2 + i/√two = ii/√2 = √2

How can you evaluate the value of cos 45?

Nosotros tin utilise Pythagoras theorem to discover the value of cos 45. Since, for a right angle, if the next angles are 45 degrees, then the adjacent side and opposite side will be equal. Thus, we can find the value of cos 45 equal to 1/√2.

Cos 45 Cos 15

Source: https://byjus.com/maths/cos-45-degrees/

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