Worked Problem · Algebra II / Trigonometry
If tan θ = −√6 in Quadrant II, What Is cos θ? A Step-by-Step Solution
A worked Quadrant II trig identity problem — the kind that shows up on the NY Algebra II Regents, SAT math, and AP Pre-Calculus exams — solved the way we teach it at SOMATH, math tutoring on the Upper West Side.
Direct answer: If tan θ = −√6 and θ terminates in Quadrant II, then cos θ = −1/√7 (equivalently, −√7 ⁄ 7 in rationalized form). The fastest path uses the Pythagorean identity 1 + tan²θ = sec²θ. The whole solution is four short lines, and the only place students lose the point is the sign — so we'll spend most of the time on that.
The problem (Question 22)
The card above is from our SHSAT/Regents review deck. Reproduced as text:
22. The tangent of an angle in standard position that terminates in Quadrant II is −√6. What is the cosine of this angle?
(1) 1/√6 (2) −√7 (3) −1/37 (4) −1/√7
Correct answer: (4) −1/√7.
Step 1 — Set up with the Pythagorean identity
The shortest route from tan θ to cos θ is the identity
1 + tan²θ = sec²θ
You can derive it in one line from sin²θ + cos²θ = 1 by dividing both sides by cos²θ, but it's worth memorizing because it converts tangent directly into secant without forcing you to draw a reference triangle.
Step 2 — Plug in tan θ = −√6
Substitute and square carefully. The negative sign disappears in the squaring step — that's why we need the Quadrant II information later.
1 + (−√6)² = sec²θ
1 + 6 = sec²θ
sec²θ = 7
Step 3 — Take the square root (and pick the right sign)
Taking the square root gives two candidates:
sec θ = ±√7
Here is the move that decides the problem. In Quadrant II, cosine is negative, so secant (its reciprocal) is also negative. Therefore:
sec θ = −√7
If a student is going to miss this problem, this is the line where it happens. Always commit the Quadrant II → cosine negative fact to memory.
Step 4 — Convert secant back to cosine
Cosine is the reciprocal of secant:
cos θ = 1 ⁄ sec θ
cos θ = 1 ⁄ (−√7)
cos θ = −1/√7
That matches choice (4). Done.
The quadrant sign chart, in one line
Most students who get this kind of problem wrong have memorized the sign chart wrong. Here it is the way we teach it at SOMATH — it fits on a single line:
- Quadrant I: all positive (sin +, cos +, tan +)
- Quadrant II: only sine positive (sin +, cos −, tan −)
- Quadrant III: only tangent positive (sin −, cos −, tan +)
- Quadrant IV: only cosine positive (sin −, cos +, tan −)
The mnemonic students reach for is "All Students Take Calculus" (A-S-T-C) starting in Quadrant I going counterclockwise — A = All, S = Sine only, T = Tangent only, C = Cosine only. Either works. The key is that in Quadrant II, cosine is negative — which is precisely what forces sec θ = −√7 in this problem.
Why the wrong answers are tempting
The distractors on this question were chosen well. Each one corresponds to a specific student error — recognizing those errors is half the battle in trig.
- (1) 1/√6 — the "reciprocal of tan without identity" trap. A student who confuses 1/tan θ with cos θ might land here. They are not equal; 1/tan θ = cot θ, not cosine.
- (2) −√7 — the "forgot to flip" trap. A student who correctly computes sec θ = −√7 but forgets that cosine is the reciprocal of secant lands here. This is the single most common error our students made before drilling the identity sequence.
- (3) −1/37 — the algebraic-mishap distractor. Comes from a sign-or-square error somewhere upstream (often (−√6)² mishandled as −6 instead of +6).
- (4) −1/√7 — correct.
Alternative method — the reference triangle
If you forget the identity, you can solve this from the unit-circle / reference-triangle picture. tan θ = opposite/adjacent, so build a reference triangle with opposite = √6 and adjacent = 1. By Pythagoras the hypotenuse is √(1² + (√6)²) = √7. Cosine is adjacent/hypotenuse, which gives 1/√7 — and then you apply the Quadrant II sign to land on −1/√7. Same answer, slightly more drawing.
Identity-first is faster on a timed exam. Triangle-first is more visual and easier to remember under stress. We teach students both and let them choose which one is their default.
Should the answer be rationalized?
Mathematically, −1/√7 and −√7 ⁄ 7 are identical. The convention is to rationalize the denominator when writing a final answer — but on this Regents-style multiple-choice question, the test writer left the answer in the un-rationalized form, so that's what you select. Always match the form used in the answer choices.
Where this problem comes from on actual NYC exams
- NY Algebra II Regents: Quadrant-aware identity problems show up almost every administration. Memorize the three identities (Pythagorean, reciprocal, quotient) cold.
- NY Algebra II / Trig honors finals (Stuyvesant, Bronx Science, Hunter, Beacon, LaGuardia, etc.): Expect them in any unit test on circular functions or analytic trig.
- SAT Math: The SAT tests lighter trig — basic ratios, the Pythagorean identity, and special angles. A direct Quadrant II identity problem is on the edge of typical SAT difficulty but does appear.
- AP Pre-Calculus & AP Calculus AB / BC: Assumes you can do this in under 60 seconds without thinking. Fluency at this level is a prerequisite for derivatives of trig functions.
- SHSAT: Does not test trig identities — but does test signed arithmetic and the squaring move that catches students here.
How we teach this at SOMATH (Math Tutoring UWS)
If you've read this far, the question becomes practical: what does actual mastery look like, and how do you get there? Here's the SOMATH approach to trig identity problems, in three steps:
- Memorize the three identity families cold. Pythagorean (sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = csc²θ), reciprocal (sec = 1/cos, csc = 1/sin, cot = 1/tan), and quotient (tan = sin/cos, cot = cos/sin). These should be reflexes, not lookups.
- Internalize the quadrant sign chart. Pick a mnemonic (A-S-T-C, or "All Students Take Calculus") and drill until you can name the signs of all six trig functions in any quadrant in three seconds.
- Drill mixed-quadrant problems weekly. Knowing each piece individually is necessary but not sufficient. Students lose the most points on integrated problems that combine an identity step with a sign step. We use a 10-problem mixed set every week through trig units.
That's it. No magic. If your child is missing points on Quadrant II / III / IV identity problems on their Algebra II tests, the fix is almost always one of those three buckets — and usually it's #2 (the sign chart) or #3 (mixed-quadrant drilling), not the identities themselves.
Need math tutoring on the Upper West Side?
School of Math (SOMATH) is a small-group math tutoring program at 226 W 79th St — Algebra I, Algebra II, Geometry, Pre-Calculus, AP Calculus AB / BC, SAT math, SHSAT, and Regents prep. We start every student with a free 30-minute evaluation and deliver a written diagnostic within 48 hours, even if you don't enroll.
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FAQ
If tan θ = −√6 and θ is in Quadrant II, what is cos θ?
cos θ = −1/√7. Use 1 + tan²θ = sec²θ to get sec²θ = 7, so sec θ = ±√7. In Quadrant II cosine (and therefore secant) is negative, so sec θ = −√7 and cos θ = 1/sec θ = −1/√7.
Why is cosine negative in Quadrant II?
On the unit circle, the x-coordinate of the point on the terminal side of θ equals cos θ. In Quadrant II that x-coordinate is to the left of the y-axis, so it's negative. Sine is positive there, tangent is negative, and secant is negative.
Do I have to rationalize −1/√7?
On most standardized tests, leaving the answer as −1/√7 is acceptable when that exact form is one of the listed choices. The rationalized equivalent is −√7 ⁄ 7. They are equal — pick whichever the answer choices use.
What identity should I memorize for this kind of problem?
1 + tan²θ = sec²θ. It is the fastest path from tangent to cosine without drawing a triangle. The companion identity sin²θ + cos²θ = 1 is also worth knowing cold. Both come straight from the Pythagorean theorem applied on the unit circle.
Where does this problem show up — Regents, SAT, SHSAT?
Quadrant-aware trig identities are a recurring topic on the NY Algebra II / Trigonometry Regents and on Algebra II honors finals at the NYC specialized high schools. The SAT touches lighter trig. The SHSAT does not test trig identities. AP Pre-Calculus and AP Calculus assume fluency with everything on this page.
Do you offer math tutoring on the Upper West Side specifically?
Yes — our entire program operates from 226 W 79th St on the Upper West Side (between Broadway and Amsterdam). Most of our families are within a 15-minute walk; we also have students who travel from the Upper East Side, Morningside Heights, Lincoln Square, and Hell's Kitchen. Math tutoring UWS is the core of what we do.