Cos 45 Degrees

The value of cos 45° is equal to 1/√2.

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, then the values depend on sides of the triangle. Cos of angle is equal to the ratio of the adjacent side and hypotenuse. Cos θ = Adjacent Side/Hypotenuse

Cos 45° Value

The exact value of cos 45 degrees is 1/√2 (in surd form), which is also equal to sin 45 degrees. It is an irrational number, equal to 0.7071067812… in decimal form. The approximate value of cos 45 is equal to 0.7071.

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

Proof

Suppose we have a right-angled triangle, in which the other two angles are equal to 45 degrees. Now, if the angles of a right triangle are 45 degrees, then the adjacent sides are equal in length.

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

According to the Pythagoras theorem, we know that, Hypotenuse²= Perpendicular² + Adjacent Side²

PQ² = PR² + QR²

⇒ r² = l² + l²

⇒ r = √2 l

⇒ l/r = 1/√2

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

Therefore, we 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/√2

Also, by trigonometric identities, we know that,

Sin² x + cos² x = 1

Or Cos² x = 1 – sin² x

Put x = 45°

Cos² 45° = 1 – sin² 45°

Put the value of sin 45° in the above equation.

Cos² 45° = 1 – (1/√2)²

Cos² 45° = 1 – ½

Cos 45° = √1/2 = 1/√2

Hence, we got the value of cos 45°.

Cos Ratio Table

Solved Examples

Question 1: Find the value 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 cos 45° = 1/√2

Therefore, ⇒ 2 (√3/2) – 4 (1/√2)

⇒ √3 – 2√2

Question 3: Find Cos 45 + Cos 90.

Solution: Cos 45° = 1/√2 Cos 90° = 0

Therefore, Cos 45° + cos 90° = 1/√2

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°) = ?