Learning goal and prerequisites
Recover an unknown trigonometric value while keeping its sign. Before starting, review inverse trig.
Method and conditions
Use sin²θ+cos²θ=1 to find a magnitude, then select the sign from the given quadrant. Define secθ=1/cosθ, cscθ=1/sinθ and cotθ=cosθ/sinθ only where their denominators are nonzero.
Worked example: show the decisions
If sinθ=5/13 in quadrant II, cos²θ=1−25/169=144/169. Quadrant II makes cosine negative, so cosθ=−12/13 and secθ=−13/12. The positive square root alone would lose quadrant information.
Contrast: why a plausible shortcut fails
The identity 1+tan²θ=sec²θ determines the square of secant. It does not force secant itself to be positive. A reciprocal reverses numerator and denominator, not the sign.
Transfer: try before revealing the answer
If tanθ=−3/4 and θ is in quadrant IV, find secθ.
Transfer solution and check
A 3–4–5 reference triangle gives cosθ=4/5, positive in quadrant IV. Hence secθ=5/4. Squaring first gives sec²θ=25/16 and the quadrant selects the positive root.
Combine two ratios after choosing the quadrant
With sin θ=5/13 in quadrant II, cosine is −12/13. Therefore tan θ=−5/12 and cot θ=−12/5. Their sum is −25/60−144/60=−169/60. This requires both the quadrant sign and a common denominator; sine plus cosine is a different quantity.
sin(θ)=3/5 and θ lies in quadrant II. What is tan(θ)+cot(θ)?
- Recover the signed cosine, form sine/cosine and cosine/sine, then add the two ratios.
- Using these quantities: (3/5)/(-√(5^2-3^2)/5)+(-√(5^2-3^2)/5)/(3/5) = -25/12.
- Check the original domain, requested quantity and units; do not substitute a related parameter.
Compare the other answer choices
Choose positive cosine despite quadrant II, making both ratios positive.
Return tangent alone and omit the cotangent term.
Add sine and cosine instead of their two quotient ratios.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
AP Precalculus Course at a Glance