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Lesson: Proving Trigonometric Identities

Worksheet • 3 Questions

Q1:

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) β‰  sum of the areas of the squares on the legs ( 1 6 + 9 )

Q2:

Consider the identity s i n c o s 2 2 πœƒ + πœƒ = 1 . We can use this to derive two new identities.

First, divide both sides of the identity by s i n 2 πœƒ to find an identity in terms of c o t πœƒ and c o s e c πœƒ .

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

Now, divide both sides of the identity through by c o s 2 πœƒ to find an identity in terms of t a n πœƒ and s e c πœƒ .

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

Q3:

The figure shows a unit circle and a radius with the lengths of its π‘₯ - and 𝑦 -components. Use the Pythagorean theorem to derive an identity connecting the lengths 1, c o s πœƒ , and s i n πœƒ .

  • A s i n c o s 2 2 πœƒ + πœƒ = 1
  • B 1 + πœƒ = πœƒ c o s s i n
  • C s i n c o s πœƒ + πœƒ = 1
  • D 1 + πœƒ = πœƒ c o s s i n 2 2
  • E s i n c o s 2 2 πœƒ βˆ’ πœƒ = 1
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