Pythagorean Identity
Question
Answer choices
- 1
- 2
- cos²θ
- 0
- sin²θ
A Correct answer: A) 1
Set up the problem carefully before computing. sin² + cos² = 1 always. The other Pythagorean identities (1 + tan² = sec², 1 + cot² = csc²) follow from dividing this one by cos² or sin².
Carry the arithmetic through one step at a time, double-checking signs and units. The ACT writes wrong answers to catch the most common slips, so an answer that "looks right" without verification is risky.
Plug the answer back into the original expression as a sanity check whenever the problem allows. If the substitution does not balance, you have made an arithmetic mistake earlier in the work.
The underlying rule
The Pythagorean identity: sin²(θ) + cos²(θ) = 1 for every angle θ. This comes from the unit circle, where any point is (cos θ, sin θ) and lies on a circle of radius 1.
Why each wrong answer is wrong
- B) 2: A common arithmetic or sign error leads to this value; carefully redo the calculation.
- C) cos²θ: A common arithmetic or sign error leads to this value; carefully redo the calculation.
- D) 0: A common arithmetic or sign error leads to this value; carefully redo the calculation.
- E) sin²θ: A common arithmetic or sign error leads to this value; carefully redo the calculation.
Study tip
sin²θ + cos²θ = 1 is the most-tested trig identity on the ACT. Know it cold.
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