Cos 45 Degrees

The sine, cosine, and tangent are the three main ratios of trigonometry. When calculating the trigonometric ratio of any angle for a right-angled triangle, the values are dependent on the triangle's sides. The ratio of the adjacent side to the hypotenuse is equal to the cos of angle.

The cosine of angle 45 degrees is written mathematically as cos 45° in the sexagesimal method. In this article, we will learn what is cos 45 and what is the value of cos 45 degrees infraction with a few solved examples.

The exact value of cos 45 degrees is \[\frac{1}{\sqrt{2}}\] (in surd form), it is also similar to sin 45 degrees. It is an irrational number equal to 0.7071067812… in decimal form. The approximate cos 45 value is equal to 0.7071.

Therefore, 0.7071 or \[\frac{1}{\sqrt{2}}\] is a value of a trigonometric function or trigonometric ratio of standard angle (\[45^{\circ}\]).

What is the value of \[\text{Cos}45^{\circ}\]?

In trigonometry \[\text{Cos}45^{\circ}\] value  = \[Cos\frac{\pi}{4} = \frac{1}{\sqrt{2}}\].Hence, the exact value in fraction form is one by square root of two.

Proof of \[\text{Cos}45^{\circ}\]

Let's say we have a right-angled triangle with two other angles of 45 degrees each. The adjacent sides of a right triangle are equal in length if the angles are 45 degrees.

(Image Will be Updated Soon)

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

According to the Pythagoras theorem, we know that, 

\[(\text{Hypotenuse})^{2} = (\text{Perpendicular})^{2} + (\text{Adjacent Side})^{2}\]

\[(PQ)^{2} = (PR)^{2} + (QR)^{2}\]

\[ \Rightarrow(r)^{2} = (l)^{2} + (l)^{2}\]

\[ \Rightarrow~(r)^{2} = 2l^{2}\]

\[ \Rightarrow~r~= \sqrt{2l}\]

\[ \Rightarrow\frac{2}{r}= \frac{1}{\sqrt{2}}\]

Thus, \[ \frac{\text{Length of adjacent side}}{\text{Hypotenuse}} = \frac{1}{\sqrt{2}}\]

Therefore, we can say, \[\text{Cos}45^{\circ} = \frac{1}{\sqrt{2}}\]

Hence, proved.

Another method to prove \[\text{Cos}45^{\circ}\] value

We can also prove the value of \[\text{Cos45}^{\circ}\] with a trigonometry approach.

As we know, \[\text{sin}45^{\circ} = \frac{1}{\sqrt{2}}\]

Also, by trigonometric identities, we know that,

\[sin^{2}x~+~cos^{2}x = 1\] or \[\text{cos}^{2}x = 1~-~sin^{2}x\]

Put \[x = 45^{\circ}\]

\[\text{cos}^{2}45^{\circ} = 1~-~\text{sin}^{2}45^{\circ}\]

Put the value of \[\text{sin}45^{\circ}\] in the above equation.

\[\text{Cos}^{2} 45^{\circ} = 1~-~\left (\frac{1}{\sqrt{2}}\right )^{2}\]

\[\text{Cos}^{2} 45^{\circ} = 1~-~\frac{1}{2}\]

\[\text{Cos}45^{\circ} = \frac{1}{\sqrt{2}}\]

Hence, \[\text{cos45}\] degrees exact value is\[\frac{1}{\sqrt{2}}\]

Trick to remember cos trigonometric value

For this, we just have to memorize the values for \[\text{Cos}~0^{\circ}\].

There is an easy way to memorize them. Shown in below table

Solved Questions:

1. Evaluate: \[2\text{sin}60^{\circ} ~-~4\text{cos}45^{\circ}\]

Sol. Since we know the value of \[\text{sin}60^{\circ} = \frac{\sqrt{3}}{2}~and~\text{cos}45^{\circ} = \frac{1}{\sqrt{2}}\]

\[2\text{sin}60^{\circ} ~-~4\text{cos}45^{\circ}\]

\[\Rightarrow 2(\frac{\sqrt{3}}{2})~-~4(\frac{1}{\sqrt{2}})\]

\[\Rightarrow \sqrt{3}~-~2\sqrt{2}\]

Hence the value of \[2\text{sin}60^{\circ} ~-~4\text{cos}45^{\circ}\] is \[\sqrt{3}~-~2\sqrt{2}\]

2. Find the value of \[\text{Cos} 45^{\circ} + \text{Cos}90^{\circ}\]?

Sol. We know the value of \[\text{Cos} 45^{\circ} = \frac{1}{\sqrt{2}}\] and \[\text{cos}90^{\circ} = 0\].

Therefore, the value of \[\text{Cos} 45^{\circ} + \text{Cos}90^{\circ} = \frac{1}{\sqrt{2}}\].

3. Find the value of \[3\text{cos} 45^{\circ} + 2\text{sin}60^{\circ}\]

Sol. We know the value of \[\text{cos}45^{\circ}\] is \[\frac{1}{\sqrt{2}}\] and \[\text{sin}60^{\circ}\] is 1.

Now put the value in \[3\text{cos} 45^{\circ} + 2\text{sin}60^{\circ}\]

\[=3 \times \frac{1}{\sqrt{2}} + 2 \times \frac{1}{\sqrt{2}}\]

\[\frac{3}{\sqrt{2}} + 1\]

\[\frac{3+\sqrt{2}}{\sqrt{2}}\]

Hence the value of \[3\text{cos} 45^{\circ} + 2\text{sin}60^{\circ}\] is \[ \frac{3+\sqrt{2}}{\sqrt{2}}\]

Conclusion

Above, we have discussed the value of \[\text{cos} 45^{\circ}\]. We have also provided how to prove the value of \[\text{cos} 45^{\circ}\] by two methods. Hence we can conclude that the approximate value of \[\text{cos} 45^{\circ}\] is equal to 0.7071.