The purpose of this activity was for us to understand where does the main Pythagorean Identity (sin²x + cos²x = 1) come from in relation to the Unit Circle and the Pythagorean Theorem. By doing so, we would be able to understand how to derive the other two remaining Pythagorean Identities from sin²x + cos²x = 1.
Before we do anything with this, we must understand what exactly is an identity. An identity is proven fact or formula that is always true. That is why the Pythagorean Theorem is also an identity because it is a proven formula that is always true.
Then, we remember that for the Unit Circle, the Pythagorean Theorem came in terms of x, y, and r instead of a, b, and c, making it x² + y² = r². Now we notice, that while the Pythagorean Identity is equal to 1, the Pythagorean Theorem that we now have is equal to r². So we'll divide both sides of the Pythagorean Theorem by r² to make one side equal to one. So it is now x²/r² + y²/r² = 1, which we can arrange into (x/r)² + (y/r)² = 1 because they still both still mean the same thing.
After that, remember how the ratio for cosine in the Unit Circle was x/r and the ratio for sine in the Unit Circle was y/r...See it yet. If cos is x/r and sin is y/r, then we can plug them into (x/r)² + (y/r)² = 1 to make it cos²θ + sin²θ = 1, which we were trying to find. That is why cos²θ + sin²θ = 1 is referred to as the Pythagorean Identity, because it is basically the Pythagorean Theorem moved around algebraically and manipulated to look like that, but it is still essentially the same thing. We can prove that it is an identity by substituting an ordered pair from the Unit Circle. Say we have a 45° angle, which has an ordered pair of (√2/2, √2/2). Sin45 would equal √2/2 and cos45 would equal √2/2. When we plug them into the identity, (√2/2)² + (√2/2)² = 1. (√2/2)² would equal 2/4 or 1/2, so 1/2 +1/2 = 1, which is true.
To derive the identity with secant and tangent by dividing sin²x + cos²x = 1 by cos²x on both sides. That becomes (y/r)²/(x/r)² + (x/r)²/(x/r)² = 1/(x/r)². When simplified, the r's will cancel out to make tangent (y/x), and the cosines will cancel to make 1. From the reciprocal identities, we know that 1/(x/r)² is equal to sec². That is how it ends up as tan²x + 1 = sec²x.
To derive the identity with cosecant and cotangent, Now instead of dividing sin²x + cos²x = 1 by cosine, we'll divide both sides by sin²x.The sines will cancel out to make 1. Cos/sin is the same as (x/r)²/(y/r)², so when it simplifies you end up with (x/y)² which is equal to cot²x because of the reciprocal identities. 1/sin²x is equal to cscx² due once again to the reciprocal identities. You end up with 1 + cot²x = csc²x.
Inquiry Activity Reflection:
1. “The connections that I see between Units N, O, P, and Q so far are…” that they all relate to the Unit Circle and trig functions in some way. They also have something to with triangles like the Pythagorean Theorem/Identity to solve for what we are looking for.
2. “If I had to describe trigonometry in THREE words, they would be…” triangles, ratios, and measures.
1. Where does sin²x + cos²x = 1 come from? (Refer to Unit Circle and Pythagorean Theorem)
Before we do anything with this, we must understand what exactly is an identity. An identity is proven fact or formula that is always true. That is why the Pythagorean Theorem is also an identity because it is a proven formula that is always true.
Then, we remember that for the Unit Circle, the Pythagorean Theorem came in terms of x, y, and r instead of a, b, and c, making it x² + y² = r². Now we notice, that while the Pythagorean Identity is equal to 1, the Pythagorean Theorem that we now have is equal to r². So we'll divide both sides of the Pythagorean Theorem by r² to make one side equal to one. So it is now x²/r² + y²/r² = 1, which we can arrange into (x/r)² + (y/r)² = 1 because they still both still mean the same thing.
After that, remember how the ratio for cosine in the Unit Circle was x/r and the ratio for sine in the Unit Circle was y/r...See it yet. If cos is x/r and sin is y/r, then we can plug them into (x/r)² + (y/r)² = 1 to make it cos²θ + sin²θ = 1, which we were trying to find. That is why cos²θ + sin²θ = 1 is referred to as the Pythagorean Identity, because it is basically the Pythagorean Theorem moved around algebraically and manipulated to look like that, but it is still essentially the same thing. We can prove that it is an identity by substituting an ordered pair from the Unit Circle. Say we have a 45° angle, which has an ordered pair of (√2/2, √2/2). Sin45 would equal √2/2 and cos45 would equal √2/2. When we plug them into the identity, (√2/2)² + (√2/2)² = 1. (√2/2)² would equal 2/4 or 1/2, so 1/2 +1/2 = 1, which is true.
2. How do we derive the other two remaining Pythagorean Identities from sin²x + cos²x = 1.
To derive the identity with secant and tangent by dividing sin²x + cos²x = 1 by cos²x on both sides. That becomes (y/r)²/(x/r)² + (x/r)²/(x/r)² = 1/(x/r)². When simplified, the r's will cancel out to make tangent (y/x), and the cosines will cancel to make 1. From the reciprocal identities, we know that 1/(x/r)² is equal to sec². That is how it ends up as tan²x + 1 = sec²x.
To derive the identity with cosecant and cotangent, Now instead of dividing sin²x + cos²x = 1 by cosine, we'll divide both sides by sin²x.The sines will cancel out to make 1. Cos/sin is the same as (x/r)²/(y/r)², so when it simplifies you end up with (x/y)² which is equal to cot²x because of the reciprocal identities. 1/sin²x is equal to cscx² due once again to the reciprocal identities. You end up with 1 + cot²x = csc²x.
Inquiry Activity Reflection:
1. “The connections that I see between Units N, O, P, and Q so far are…” that they all relate to the Unit Circle and trig functions in some way. They also have something to with triangles like the Pythagorean Theorem/Identity to solve for what we are looking for.
2. “If I had to describe trigonometry in THREE words, they would be…” triangles, ratios, and measures.
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