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A short class with Sarah and her students. The other ways are under More.
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Let's test sin²θ + cos²θ = 1 at θ = 30°. What are sin 30° and cos 30°?
sin 30° is 1/2 and cos 30° is √3/2.
Squaring gives 1/4 and 3/4, and they add up to 1.
It works. Have we proved the identity?
We've only shown it for 30°. It might fail for some other angle.
Exactly. A check is useful, but a proof must work for every angle at once.
Take a right triangle with opposite side a, adjacent side b and hypotenuse c. Write sin θ and cos θ.
sin θ = a/c and cos θ = b/c.
Now square them and add.
a²/c² + b²/c², which is (a² + b²)/c². Pythagoras says a² + b² = c², so it's 1.
That covers acute angles. How about 150°?
Use the unit circle. The point (cos θ, sin θ) is 1 unit from the origin, so x² + y² = 1 for any angle.
If sin θ = 3/5 and θ is acute, find cos θ.
cos²θ = 1 − 9/25 = 16/25, so cos θ = 4/5.
What if θ were obtuse?
Cosine is negative between 90° and 180°, so cos θ would be −4/5.
Now prove that (1 − cos θ)(1 + cos θ) = sin²θ.
Expand the left side: 1 − cos²θ. By the identity, that's sin²θ, which is the right side.
sin²θ + cos²θ = 1 for every angle θ, where sin²θ means (sin θ)².
substituting θ = 30° verifies the identity for one angle; a proof must work for all angles.
(a² + b²)/c² = 1 because a² + b² = c² by Pythagoras' theorem.
the point (cos θ, sin θ) lies on x² + y² = 1, so the identity holds for any angle.
sin²θ = 1 − cos²θ and cos²θ = 1 − sin²θ let you find one ratio from the other, choosing the sign from the angle.
work on one side only and transform it into the other side.
Check at θ = 30°: (1/2)² + (√3/2)² = 1/4 + 3/4 = 1.
Check at θ = 45°: (√2/2)² + (√2/2)² = 1/2 + 1/2 = 1.
If sin θ = 3/5 and θ is acute: cos²θ = 1 − 9/25 = 16/25, so cos θ = 4/5 and tan θ = 3/4.
Prove (1 − cos θ)(1 + cos θ) = sin²θ: the left side expands to 1 − cos²θ, which equals sin²θ by the identity.
Sine squared, cosine squared, add the pair; one is the answer, everywhere.
A tiny robot walks round a circle of radius 1. Wherever it stops, its east-west distance squared plus its north-south distance squared is 1, because it never leaves the circle. Those distances are cos θ and sin θ.
"Squares of the sides make the square of the slide" - the legs squared add up to the hypotenuse squared, which is why the identity works.
S.C.O. - Sine squared plus Cosine squared equals One.
Term
State the Pythagorean identity.
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What does sin²θ mean?
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Why is checking θ = 30° not a proof?
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Write cos²θ in terms of sin θ.
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If sin θ = 3/5 and θ is acute, what is cos θ?
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An equation that is true for every value of its variable, such as sin²θ + cos²θ = 1.
sin²θ + cos²θ = 1, which follows from Pythagoras' theorem.
The square of sin θ, that is (sin θ)².
The circle of radius 1 centred at the origin, with equation x² + y² = 1.
Checking that a statement holds for particular values; it supports but does not prove an identity.
A logical argument showing a statement is true in every case.
An acute angle is less than 90°; an obtuse angle is between 90° and 180°.
Example 1 is about the Pythagorean identity sin²θ + cos²θ = 1. Checking θ = 30° gives 1/4 + 3/4 = 1, but a check is not a proof; this lesson proves the identity for every angle using a right triangle and the unit circle, then uses it to find values and prove other identities.
The identity sin²θ + cos²θ = 1 holds for every angle θ. Substituting θ = 30° (sin 30° = 1/2 and cos 30° = √3/2) gives 1/4 + 3/4 = 1, which is a useful check, but testing one angle cannot prove that something is true for all angles. A proof uses Pythagoras' theorem: in a right triangle the squares of the two shorter sides add up to the square of the hypotenuse, and on the unit circle the point (cos θ, sin θ) satisfies x² + y² = 1. This lesson sets out both proofs, rearranges the identity into sin²θ = 1 − cos²θ and cos²θ = 1 − sin²θ, uses it to find cos θ from sin θ, and shows how to write a clear proof of other identities.
Notation. sin²θ means (sin θ)², the square of the sine of θ. It does not mean sin(θ²).
Checking with a special angle. For θ = 30°, sin 30° = 1/2 and cos 30° = √3/2, so sin²30° + cos²30° = 1/4 + 3/4 = 1. The same happens for 45°, (√2/2)² + (√2/2)² = 1/2 + 1/2 = 1, and for 60°, (√3/2)² + (1/2)² = 3/4 + 1/4 = 1.
These checks show the identity working, but they do not prove it. An identity claims to be true for every angle, and no number of examples can cover every angle. (One example where it failed would disprove it.) A proof has to work for a general θ.
Proof 1: the right triangle (for acute θ). Take a right triangle with angle θ, opposite side a, adjacent side b and hypotenuse c. Then sin θ = a/c and cos θ = b/c, so
sin²θ + cos²θ = a²/c² + b²/c² = (a² + b²)/c²
By Pythagoras' theorem a² + b² = c², so the expression equals c²/c² = 1.
Proof 2: the unit circle (for every θ). On a circle of radius 1 centred at the origin, the point reached by turning θ anticlockwise from the positive x-axis has coordinates (cos θ, sin θ). Every point on this circle is 1 unit from the origin, so it satisfies x² + y² = 1. Substituting x = cos θ and y = sin θ gives cos²θ + sin²θ = 1. This proof covers obtuse, reflex and negative angles too.
Rearranged forms. sin²θ = 1 − cos²θ and cos²θ = 1 − sin²θ. These are often the forms you need.
Using the identity to find values. If sin θ = 3/5 and θ is acute, then cos²θ = 1 − 9/25 = 16/25, so cos θ = 4/5 and tan θ = sin θ ÷ cos θ = 3/4. If instead θ were obtuse (between 90° and 180°), cosine is negative there, so cos θ = −4/5.
Proving other identities. Start with the more complicated side, work on that side only, and finish with exactly the other side.
A common mistake. Do not start by writing the identity you want to prove and then changing both sides until you get something true, such as 0 = 0. That assumes the very thing you are trying to prove.