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Mumz [18]
3 years ago
11

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

Mathematics
2 answers:
mash [69]3 years ago
6 0

Explanation:

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

<em>Trig functions</em> 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,

  • sin(30°) = cos(60°) = 1/2
  • sin(60°) = cos(30°) = √(3/4)
  • sin(45°) = cos(45°) = √(1/2)

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.

_____

<em>Comment on the question wording</em>

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.

Reptile [31]3 years ago
3 0

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

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