What is the Value of Cos 37° in Fraction?

In competitive physics and mathematics (especially in mechanics problems involving inclined planes and vectors), the angle $37^\circ$ appears very frequently. It is highly important to memorize its fractional value to solve problems quickly without a calculator.

Question (Click to Flip)

How is this used in physics?

Answer

When a force block is sliding down an inclined plane at an angle of 37°, you must resolve the gravity vector into its components. The horizontal component will use sin 37 (3/5) and the normal force will use cos 37 (4/5).

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Key Facts

Note that $37^\circ$ and $53^\circ$ are approximations. The exact angle is actually $36.87^\circ$, but in all high school physics exams, it is rounded to $37^\circ$ to make calculations easy.

Because $\sin 37^\circ = \cos 53^\circ$, they are complementary angles.

1. The Value

The universally accepted fractional value for $\cos 37^\circ$ is $\frac{4}{5}$. (In decimal form, it is $0.8$).

2. The Magic of the 3-4-5 Triangle

To understand why the value is $4/5$, you must look at a special right-angled triangle called the '3-4-5 Triangle'.

If a right-angled triangle has sides of length 3, 4, and a hypotenuse of 5 (which satisfies Pythagoras' theorem $3^2 + 4^2 = 5^2$), the interior angles are perfectly approximately $90^\circ, 53^\circ, \text{and } 37^\circ$.
In this triangle, the side adjacent to the $37^\circ$ angle has a length of 4.
The hypotenuse has a length of 5.
Since $\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}$, therefore $\cos 37^\circ = \frac{4}{5}$.

3. Related Important Values

Using the same 3-4-5 triangle, you should also memorize these related values:

$\sin 37^\circ = \frac{3}{5}$ (0.6)
$\tan 37^\circ = \frac{3}{4}$ (0.75)
$\cos 53^\circ = \frac{3}{5}$ (0.6)
$\sin 53^\circ = \frac{4}{5}$ (0.8)

Questions and Answers

How is this used in physics?+

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