Worksheet: Proving Trigonometric Identities

In this worksheet, we will practice deriving the Pythagorean identity and solving proofs problems related to it.

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 .

  • A s i n c o s 2 2 𝜃 𝜃 = 1
  • B s i n c o s 𝜃 + 𝜃 = 1
  • C 1 + 𝜃 = 𝜃 c o s s i n 2 2
  • D s i n c o s 2 2 𝜃 + 𝜃 = 1
  • E 1 + 𝜃 = 𝜃 c o s s i n

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 .

  • A 1 + 𝜃 = 𝜃 c o t s i n 2 2
  • B 1 + 𝜃 = 𝜃 t a n c o s e c 2 2
  • C 1 + 𝜃 = 𝜃 t a n s i n 2 2
  • D 1 + 𝜃 = 𝜃 c o t c o s e c 2 2
  • E 1 + 𝜃 = 𝜃 c o t s e c 2 2

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 .

  • A t a n s e c 2 2 𝜃 + 1 = 𝜃
  • B t a n s i n 2 2 𝜃 + 1 = 𝜃
  • C t a n c o s e c 2 2 𝜃 + 1 = 𝜃
  • D c o t s e c 2 2 𝜃 + 1 = 𝜃
  • E t a n c o s 2 2 𝜃 + 1 = 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.

  • A area of the square on the hypotenuse (25) sum of the areas of the squares on the legs ( 1 6 + 9 )
  • B area of the square on the hypotenuse (25) > sum of the areas of the squares on the legs ( 1 6 + 9 )
  • C area of the square on the hypotenuse (25) < sum of the areas of the squares on the legs ( 1 6 + 9 )
  • D area of the square on the hypotenuse (25) = sum of the areas of the squares on the legs ( 1 6 + 9 )

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