Weird Trig problem

[paradise737]

In this assignment of mine, I'm given an isosceles triangle with sides A, B, and C and <ACB = 90 degrees.

It's an isoceles triangle, so the other two angles are 45 degrees each. But then the problem says "Find the exact values of sin A, cos A, and tan A, based on geometric principles; do not use a calculator."

How do I go about answering this? According to my calculator, sin(45) and cos(45) = .707, while tan(45) = 1--great, so now what? How would I go about figuring that out without a calculator...?

 

[akoaysigod]

Since all triangles are proportional you could try assigning numbers to the sides. Two of the angles are 45 degrees which means you can assign the same numbers to the sides opposite of those angles, I would go with 1 but any number works. Use pythagereoms? theorem to find the third side, then use what you know about trig functions, soh cah toa. I never had to spell pythageosirj but I hope you know what I mean. No matter what number you pick you'll get what your calculator told you was the answer.

 

[soroban]

Hello, paradise737!

$\displaystyle \text{Given an isosceles right triangle with sides }A, B, C\,\text{ and } \angle C = 90^o.$

It's an isoceles triangle, so the other two angles are 45 degrees each . ] . . . Right!

$\displaystyle \text{Find the exact values of }\sin A, \cos A, \tan A,\text{ based on geometric principles. }\:\text{Do not use a calculator.}$

Make a sketch . . .

    B *
      | *
      |   *  h
    1 |     *
      |       *
      |      45 *
    C * - - - - - * A
            1

You know it is an isosceles right triangle.
Let the two equal sides equal 1.

$\displaystyle \text{From Pythagorus, we hve: }\;1^2+1^2 \:=\:h^2$

. . $\displaystyle \text{Hence: }\;h^2 \:=\:2 \quad\Rightarrow\quad h = \sqrt{2}$

$\displaystyle \text{We have a }45^o\text{ angle with: }\;\begin{Bmatrix}opp &=& 1 \\ adj &=& 1 \\ hyp &=& \sqrt{2} \end{Bmatrix}$


\(\displaystyle \text{Now you can find }\sin45^o,\:\cos45^o,\:\tan45^o \;\hdots \text{ right?}\)

 

[Mrspi]

Since all triangles are proportional you could try assigning numbers to the sides. <-----I DON"T THINK SO

Any two triangles with the same angle measurements are similar.

Any two 45-45-90 triangles, for example, are similar so the corresponding sides are in the same ratio.

Draw a triangle with one right angle and two 45-degree angles. Because this triangle has two equal angles, the sides opposite these angles, which are the legs of the right triangle, must have the same length.

You can choose any number to represent the length of one of these legs. The other leg must have the same length.

Suppose you choose 2 for the length of one leg. Then the other leg has length 2 also.

And Pythagoras tells us that in any right triangle,

leg[sup:17fkulnj]2[/sup:17fkulnj] + leg[sup:17fkulnj]2[/sup:17fkulnj] = hypotenuse[sup:17fkulnj]2[/sup:17fkulnj]

Let h = length of the hypotenuse. And we already know that in our particular right triangle, each leg has a length of 2.

2[sup:17fkulnj]2[/sup:17fkulnj] + 2[sup:17fkulnj]2[/sup:17fkulnj] = h[sup:17fkulnj]2[/sup:17fkulnj]

4 + 4 = h[sup:17fkulnj]2[/sup:17fkulnj]

8 = h[sup:17fkulnj]2[/sup:17fkulnj]

Take the square root of both sides to find h....

sqrt(8) = sqrt(h[sup:17fkulnj]2[/sup:17fkulnj])

sqrt(8) = h

Simplify the radical on the left side. sqrt(8) = sqrt(4)*sqrt(2), or 2 sqrt(2)

2 sqrt (2) = h

Ok...now you know all of the sides of the right triangle we've created. To find the trig functions for 45 degrees, pick one of the 45-degree angles, and use these definitions:

sin 45 = opposite leg / hypotenuse

cos 45 = adjacent leg / hypotenuse

tan 45 = opposite leg / adjacent leg

For example, to find sin 45, look at our triangle. The opposite leg has length 2 . The hypotenuse has length 2 sqrt(2). So,
sin 45 = 2 / 2 sqrt(2), or 1/sqrt(2)

 

[akoaysigod]

Sorry, my choice of words was poor but the conclusion much the same, as far as this problem goes. I'll try to be more correct next time.