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All Flashcards in Topic 3.9
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3.9.18 cards
Define sec θ, csc θ and cot θ.
sec θ = 1/cos θ, csc θ = 1/sin θ, cot θ = 1/tan θ = cos θ/sin θ.
Which basic ratio does SECANT pair with?
Cosine — sec θ = 1/cos θ (match the third letter: se-C-ant ↔ -C-osine).
State the identity linking tan and sec.
1 + tan²θ = sec²θ (divide sin²+cos²=1 by cos²θ).
State the identity linking cot and csc.
1 + cot²θ = csc²θ (divide sin²+cos²=1 by sin²θ).
How do you solve an equation containing sec x?
Rewrite sec x = 1/cos x, take reciprocals to get cos x = …, then solve as a normal cosine equation.
Where is sec θ undefined?
Wherever cos θ = 0, i.e. θ = 90°, 270°, … (π/2, 3π/2, …).
If csc θ = 13/12 in Q1, find cot θ.
1 + cot²θ = (13/12)² = 169/144 ⇒ cot²θ = 25/144 ⇒ cot θ = +5/12 (Q1).
Find sec(π/3).
1/cos(π/3) = 1/(1/2) = 2.
3.9.28 cards
Why does sine need a restricted domain to have an inverse?
Over all reals sine repeats, so sin x = c has many solutions. Restricting to [−π/2, π/2] makes it one-to-one, so it can be reversed.
Domain and range of arcsin x?
Domain [−1, 1], range [−π/2, π/2].
Domain and range of arccos x?
Domain [−1, 1], range [0, π].
Domain and range of arctan x?
Domain all real numbers, range (−π/2, π/2) (open).
Exact value of arctan(√3)?
π/3, since tan(π/3) = √3 and π/3 is in (−π/2, π/2).
Exact value of arccos(−1/2)?
2π/3 (cosine is −1/2 there, and 2π/3 is in [0, π]).
Simplify cos(arcsin x).
Let θ = arcsin x ⇒ sin θ = x; cos θ = √(1 − x²) (non-negative on [−π/2, π/2]).
How do you sketch y = arcsin x from y = sin x?
Take the rising piece of sine on [−π/2, π/2] and reflect it in the line y = x.
Topic 3.9 study notes
Full notes & explanations for Reciprocal & inverse trig (HL only)
Math AA exam skills
Paper structures, command terms & tips
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