Why isn't cos(22.5°) = 2/√5?

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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?

phinds
Gold Member
2019 Award
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.

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:
lurflurf
Homework Helper
The relation between angle and slope is

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

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.