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Thanks to Amit for the suggestion! Which quantity is greater? Calculators are not allowed.

x = (sin 1°)/(sin 2°)
y = (sin 3°)/(sin 4°)

As usual, watch the video for a solution.

Comparing Sine Ratios Without A Calculator

Or keep reading.
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“All will be well if you use your mind for your decisions, and mind only your decisions.” Since 2007, I have devoted my life to sharing the joy of game theory and mathematics. MindYourDecisions now has over 1,000 free articles with no ads thanks to community support! Help out and get early access to posts with a pledge on Patreon.

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Answer To Which Sine Ratio Is Greater

(Pretty much all posts are transcribed quickly after I make the videos for them–please let me know if there are any typos/errors and I will correct them, thanks).

Method 1: product to sum

x = (sin 1°)/(sin 2°)
y = (sin 3°)/(sin 4°)

Since sin 2° and sin 4° are greater than 0, we can multiply both numbers by them, so we are equivalently comparing:

(sin 2°)(sin 4°)x = (sin 4°)(sin 1°)
(sin 2°)(sin 4°)y = (sin 3°)(sin 2°)

So we can compare the product of sines, and whichever is larger depends only on whether x or y is larger. We now use the product to sum formula:

sin A sin B = 0.5(cos(AB) – cos(A + B))

(sin 4°)(sin 1°)
= 0.5(cos(4° – 1°) – cos(4° + 1°))
= 0.5(cos 3° – cos 5°)

(sin 3°)(sin 2°)
= 0.5(cos(3° – 2°) – cos(3° + 2°))
= 0.5(cos 1° – cos 5°)

We now want to compare the final two expressions. Since 0.5 is greater than 0 we can cancel it, and we can cancel subtracting cos(5°) from both sides. Thus we are left to compare:

2(sin 2°)(sin 4°)x + cos 5° = cos 3°
2(sin 2°)(sin 4°)y + cos 5° = cos 1°

Since cosine is a decreasing function between 0 and 90 degrees, cos 3° is less than cos 1°. Therefore we must have x is less than y, so

x = (sin 1°)/(sin 2°)
is less than
y = (sin 3°)/(sin 4°)

Method 2: small angle approximation

The Indian mathematician Madhava described this series for sine hundreds of years before European mathematicians.

sin x = xx3/3! + x5/5! + …

Note x is in radians. When x is a small value close to 0, the cubic and subsequent terms will be very small and equal to approximately 0. So we have the approximation:

sin x ≈ x

We are dealing with degree values that are small, so their radian values will also be small. So we convert the angles and then use the small angle approximation:

(sin 1°)/(sin 2°)
= (sin (π/180))/(sin (2π/180))
≈ (π/180)/(2π/180)
= 1/2

(sin 3°)/(sin 4°)
= (sin (3π/180))/(sin (4π/180))
≈ (3π/180)/(4π/180)
= 3/4

Clearly 1/2 is less than 3/4, so we have:

x = (sin 1°)/(sin 2°)
is less than
y = (sin 3°)/(sin 4°)

References

Doubtnut solution by sine sum of angles
https://www.doubtnut.com/question-answer/which-is-greater-sin1-sin2-or-sin3-sin4-121887217

Madhava
https://en.wikipedia.org/wiki/Madhava_of_Sangamagrama

Archer in preview picture
https://en.wikipedia.org/wiki/Longbow#/media/File:Crecy_village_sign.JPG
By Peter Lucas – Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=19215883
https://creativecommons.org/licenses/by-sa/3.0/deed.en

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