Difference between revisions of "Pythagorean identities"
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<math>\sin^2x + \cos^2x = 1</math> | <math>\sin^2x + \cos^2x = 1</math> | ||
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<math>1 + \cot^2x = \csc^2x</math> | <math>1 + \cot^2x = \csc^2x</math> | ||
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<math>\tan^2x + 1 = \sec^2x</math> | <math>\tan^2x + 1 = \sec^2x</math> | ||
| − | Using the unit circle definition of trigonometry, because the point <math>(\cos (x), \sin (x))</math> is defined to be on the unit circle, it is a distance one away from the origin. Then by the distance formula, <math>\sin^2x + \cos^2x = 1</math>. To derive the other two Pythagorean identities, divide by either <math>\sin^2 (x)</math> or <math>\cos^2 (x)</math> and substitute the respective trigonometry in place of the ratios to obtain the desired result. | + | |
| + | Using the unit circle definition of trigonometry, because the point <math>(\cos (x), \sin (x))</math> is defined to be on the unit circle, it is a distance one away from the origin. Then by the distance formula, <math>\sin^2x + \cos^2x = 1</math>. | ||
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| + | Another way to think of it is as follows: Suppose that there is a right triangle <math>ABC</math> with the right angle at <math>B</math>. Then, we have: | ||
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| + | <math>BC^2+AB^2=AC^2</math> | ||
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| + | <math>\frac{BC^2}{AC^2}+\frac{AB^2}{AC^2}=\frac{AC^2}{AC^2}</math> | ||
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| + | <math>\sin^2A+cos^2A=1</math>. | ||
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| + | To derive the other two Pythagorean identities, divide <math>\sin^2A+cos^2A+1</math> by either <math>\sin^2 (x)</math> or <math>\cos^2 (x)</math> and substitute the respective trigonometry in place of the ratios to obtain the desired result. | ||
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Latest revision as of 13:07, 3 January 2024
The Pythagorean identities state that
Using the unit circle definition of trigonometry, because the point 
is defined to be on the unit circle, it is a distance one away from the origin. Then by the distance formula, .
Another way to think of it is as follows: Suppose that there is a right triangle 
with the right angle at 
. Then, we have:
.
To derive the other two Pythagorean identities, divide 
by either 
or 
and substitute the respective trigonometry in place of the ratios to obtain the desired result.
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