# Why isn't cos(22.5°) = 2/√5?

bonodut
I've never really learned any trigonometry and I'm wondering if someone could tell me where I'm going wrong here.

My reasoning is this:

The hypotenuse of a 1,1,√2 right triangle is at a 45° angle to its base.

Halving that angle should require that the base leg be doubled if the height leg is kept at 1, giving the new triangle lengths of 1,2,√5

The cosine is adjacent/hypotenuse which would be 2/√5, yet calculators give a different answer. Why is this?

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Halving that angle should require that the base leg be doubled if the height leg is kept at 1, giving the new triangle lengths of 1,2,√5
Your mistake is in thinking that a 1, 2, sqrt(5) triangle has one angle 22.5 degrees.

bonodut
Your mistake is in thinking that a 1, 2, sqrt(5) triangle has one angle 22.5 degrees.
Halving that angle should require that the base leg be doubled if the height leg is kept at 1

So this assumption isn't correct?

Edit: I see now that it's not but I don't understand why... Seems like it should be

2nd edit: my problem was that I was equating angle with slope. I like math... Too bad I'm so bad at it! :/

Last edited:
Homework Helper
The relation between angle and slope is

$$\mathrm{slope}=\dfrac{\mathrm{rise}}{\mathrm{run}}=\dfrac{\Delta y}{\Delta x}=\dfrac{\sin(\theta)}{\cos(\theta)}=\tan(\theta)$$

ellipsis
Raghav Gupta
So this assumption isn't correct?

Edit: I see now that it's not but I don't understand why... Seems like it should be

2nd edit: my problem was that I was equating angle with slope. I like math... Too bad I'm so bad at it! :/
Draw a right angle triangle using protractor,scale,pen or pencil which I just did.By drawing the dimensions of right angle triangle of sides 1,2,## \sqrt 5 ## you will see that the angle made by hypotenuse and base is approximately 26.5° not your 22.5°.So, cos 26.5°= 2/##\sqrt 5## So the calculators were right and you may have made a minute mistake.