Maths: Finding the Value of sin18°∙sin54° without Calculator

Let's find the value of sin 18°∙sin 54° without a calculator. Well, assuming the calculator has the "sin" button on it. That type of calculator. Scientific calculator.

Hang on. 🤔

Indeed!

"Calculator", BY DEFAULT, has no sin / cos / tan button on it! That's specifically a scientific calculator.

"Calculator" (default): + − × ÷

‼️

As such:

I can solve sin18°∙sin54° WITH a calculator! HAHAHA.

(Looks at the calculator.) Where's the SIN BUTTON?!

Oh, well. Too late.

So instead, we use a laptop.

Jest!

Right.

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Before we start, here's the reference.

• The double-angle formula for sine function is: sin 2α = 2∙sin α∙cos α
• The co-function (0° ≤ α ≤ 90°) is: sin α = cos (90° - α) Or: cos α = sin (90° - α)

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Sum of Angles

This is a brain teaser for sum of angles in the trigonometric function.

So.

To solve sin 18°∙sin 54°, we can MAGICALLY start from sin 72°. Indeed.

Why 72°? 🤔

Because 18° + 54° = 72°

And 72 ÷ 2 = 36

72 is 36 being doubled. ✨

It FITS our theme, DOUBLE-ANGLE! ✅

The magic is revealed. 🪄

🧙‍♂️

We will see how the addition (18° + 54°) and the double-angle tool will help us solve the sin18°∙sin54° without using a calculator.

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The Steps

This is the image of the steps:

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The answer is ¼ or 0.25.

This method only works for 18° and 54° — both sine functions.

sin18°∙sin54° using co-function is equivalent with sin18°∙cos36° or cos72°∙sin54° or cos72°∙cos36°.

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Demonstration

This is an application for this particular post, for both sine functions, so we can prove the result from the steps above.

The output is rounded up to two decimal places.

We can either hit Enter on the keyboard or 🖱️ SUBMIT button after filling the number inputs.

sin °∙ sin °

This employs JavaScript's —

• Math.sin() for the calculation,
• and Math.PI for conversion from radian to degree.

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Submit

Let's derail a bit.

You know, the term "submit" in modern world. Submit, submission, and the lot.

Back then, "submit" meant —

accept or yield to a superior force or to the authority or will of another person.

Well, I suppose until now — the original meaning. Broadened to... modern meaning. Especially in computer world. We've seen it, right? PLENTY of times! SUBMIT this, SUBMIT THAT! Even my application above does it.

In judicial context, it still makes sense. But in digital realm, it is rather comical.

Submit. Now. Or else.

👀 Oui, millourt. 🙇🙏