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In this lesson, we will learn how to derive the Pythagorean identities and solve proofs problems.
Q1:
The figure shows a unit circle and a radius with the lengths of its 𝑥 - and 𝑦 -components. Use the Pythagorean theorem to derive an identity connecting the lengths 1, c o s 𝜃 , and s i n 𝜃 .
Q2:
Consider the identity s i n c o s 2 2 𝜃 + 𝜃 = 1 . We can use this to derive two new identities.
First, divide both sides of the identity by s i n 2 𝜃 to find an identity in terms of c o t 𝜃 and c o s e c 𝜃 .
Now, divide both sides of the identity through by c o s 2 𝜃 to find an identity in terms of t a n 𝜃 and s e c 𝜃 .
Q3:
The lengths of the sides of the right triangle shown in the figure are 3, 4, and 5. Find the areas of the squares on the three sides, and find a relationship between them.
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