How can the Pythagorean theorem be used to find the exact values of certain "special angles"?

In a 45-45-90 triangle, the two legs are congruent. Let's call them x. The hypotenuse is equal to 1 as we're using the unit circle. The hypotenuse of the triangle is the same as the radius of the unit circle.

a = x

b = x

c = 1

Use those values in the Pythagorean theorem to solve for x.

a^2 + b^2 = c^2

x^2 + x^2 = 1^2

2x^2 = 1

x^2 = 1/2

x = sqrt( 1/2 )

x = sqrt(1)/sqrt(2)

x = 1/sqrt(2)

x = sqrt(2)/2 ... rationalizing the denominator

So this right triangle has legs that are sqrt(2)/2 units long. Once we know the legs of the triangle, we can divide them over the hypotenuse to find the sine and cosine values.

sin(angle) = opposite/hypotenuse

sin(45) = (sqrt(2)/2) / 1

sin(45) = sqrt(2)/2

and

cos(angle) = adjacent/hypotenuse

cos(45) = (sqrt(2)/2) / 1

cos(45) = sqrt(2)/2

For a 30-60-90 triangle, we would have

a = 1

b = x

c = 2

so,

a^2+b^2 = c^2

1^2+x^2 = 2^2

1+x^2 = 4

x^2 = 4-1

x^2 = 3

x = sqrt(3)

The missing leg is sqrt(3) units long.

Once we know the three sides of the 30-60-90 triangle, you should be able to see that

sin(30) = 1/2

sin(60) = sqrt(3)/2

cos(30) = sqrt(3)/2

cos(60) = 1/2

Explanation:

The Pythagorean theorem is a theorem that can be used to find side lengths of a right triangle. That is all.

Trig functions are used to relate ratios of side lengths to the values of angles. The Pythagorean theorem may be helpful for finding the value of a trig function, and that, in turn, may be compared to the trig function of a "special angle." Hence, finding the exact value of some angles may be facilitated by use of the Pythagorean theorem.

What makes an angle "special" is that its exact trig function values are easily found rational numbers or the square root of rational numbers. For example,

30°, 45°, and 60° are considered to be "special angles", both because their exact trig function values are easily determined, and these angles appear in many regular geometric figures.

If the Pythagorean theorem reveals a triangle has side/hypotenuse ratios of 1/2 or √(1/2) or √(3/4), then the exact value of the angle in the triangle will be known.

Comment on the question wording

We find increasingly that math question wording suffers from imprecision and questionable use of the English language. Here the question is asking how a theorem that finds lengths can be used to find the exact values of specific angles. Those angles' exact values are well-known. The Pythagorean theorem has nothing to do with finding angles, or determining the value of a special angle.

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