Question

Evaluate sin 60° without using a calculator by using ratios in a reference triangle.

Answer 1

How did you do that??? Can you explain it to me @kc_kennylau ?

Answer 2

the whole triangle is equilateral

Answer 3

He used a equilateral triangle with same length of side = 2. http://www.mathsisfun.com/definitions/equilateral-triangle.html

Answer 4

Yes, but how did you get 1, 2, and sqrt3?

Answer 5

Oops, sorry. Sent that before I caught up. Okay, so how did you know all of the sides would be 2? Or is that just a random number that got plugged in?

Answer 6

Assumed triangle has 3 sides = 2. The number 2 is chosen just because of easier to solve.

Draw the height then we have a triangle with a reference is 2, the other is 1 (due to 1 + 1 = 2). The other side, using Pythagoras theorem, \[2^2 = 1^2 + x^2 \rightarrow x = \sqrt{3}\]

Answer 7

Okay, I got all of that. But can you just pick a random number like that? I'm not trying to argue, I literally have no idea.

Answer 8

yes, you can

Answer 9

If you choose any number, it is OK.

Answer 10

So in the end, sine would be opp./hyp. That means here, it would be sqrt3/2. Is that right? It sounds wrong..

Answer 11

but that's right

Answer 12

You need to find the sin of 60 deg using a ratio of the lengths of the sides of a triangle. That means you need a right triangle that has one angle measure of 60 degrees.
To do that, you start with an equilateral triangle. The sum of the measures of the angles in a triangle is 180 degrees. A triangle with 3 congruent sides also has 3 congruent angles. If all 3 angles have the same measure, and they add up to 180, then each angle measure is 60 degrees.

Answer 13

By drawing an equilateral triangle, you know you have 3 angles of 60 degrees.



The figure shows an equilateral triangle with all sides measuring the same, x, and all angles have a measure of 60 degrees.

Answer 14

Yes, yes. I've got that @mathstudent55 . I need confirmation on the later parts..

Answer 15

Now for convenience, we pick the length of all sides of the original equilateral triangle to be 2. That means that when we draw the perpendicular to the bottom side, we have 90 degree angles and two small sides of length 1.