Why does cos(22.5°) not equal 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?

 

[phinds]

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! :/

 

[lurflurf]

The relation between angle and slope is

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

 

[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.

 

[Raghav Gupta]

Try here http://www.mathwarehouse.com/triangle-calculator/online.php
Give $\angle B$ as 90°,side C as 2 and side B as 2.236 $\equiv \sqrt 5$ $$ \angle A $$ would be calculated as 26.5°.

 

[lurflurf]

I like this diagram.

http://www.cut-the-knot.org/pythagoras/Tangent225.shtml