ACT Preparation / Unit 14 ACT 1328 worked example, 60 to 90 seconds
Using the Pythagorean Identity
The idea
Given one trig value and the quadrant, the others follow in three steps, without finding the angle at all. First, use sin²θ + cos²θ = 1 to find the missing sine or cosine. Second, choose its sign from the quadrant. Third, form the tangent as sine over cosine, and any reciprocal function the question asks for.
In practice
Worked example: cos θ = −5/13, with θ in the second quadrant. sin²θ = 1 − 25/169 = 144/169, so sin θ = ±12/13, and the second quadrant makes it positive, 12/13. Then tan θ = (12/13)/(−5/13) = −12/5, and sec θ = −13/5.
Where people slip
A right-triangle sketch gives the sizes, 5, 12 and 13, but not the signs. Only the quadrant supplies those, which is why the second step cannot be skipped.
Your place on the trail
Every lesson rests on earlier ones and opens later ones. Tap any to jump.
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