Identities & double angles
Practice Flashcards
State the Pythagorean identity.
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All Flashcards in Topic 3.6
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3.6.19 cards
State the Pythagorean identity.
sin²θ + cos²θ = 1, for every angle θ.
Rearrange for sin²θ.
sin²θ = 1 − cos²θ.
Rearrange for cos²θ.
cos²θ = 1 − sin²θ.
How do you find sin θ from cos θ?
sin θ = ±√(1 − cos²θ); pick the sign from the quadrant.
Given cos θ = 2/3 (acute), find sin θ.
sin²θ = 1 − 4/9 = 5/9 ⇒ sin θ = √5/3.
Simplify 1 − sin²θ.
cos²θ.
Simplify 1 − cos²θ.
sin²θ.
Why must you watch the sign when rooting?
√ gives only the magnitude; the quadrant decides + or −.
Key move when proving a trig identity?
Replace 1 − sin²θ or 1 − cos²θ with the other square, then cancel.
3.6.29 cards
Double-angle formula for sine?
sin 2θ = 2 sin θ cos θ (not 2 sin θ!).
Three forms of cos 2θ?
cos²θ − sin²θ = 1 − 2sin²θ = 2cos²θ − 1.
Which cos 2θ form if you only know sin θ?
1 − 2sin²θ.
Which cos 2θ form if you only know cos θ?
2cos²θ − 1.
Given sin θ = 3/5, cos θ = 4/5, find sin 2θ.
2(3/5)(4/5) = 24/25.
Given cos θ = 4/5 (acute), find cos 2θ.
2(16/25) − 1 = 7/25.
Simplify cos⁴θ − sin⁴θ.
(cos²−sin²)(cos²+sin²) = cos 2θ.
Common double-angle mistake?
Writing sin 2θ = 2 sin θ (dropping cos θ).
How do you start a double-angle exact-value problem?
Find sin θ and cos θ first (often via the Pythagorean identity), then substitute.
Topic 3.6 study notes
Full notes & explanations for Identities & double angles
Math AA SL exam skills
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