Hard Geometry and Trig

• Work top-quartile geometry and trig questions one at a time at test pace, about 105 seconds each
• Build speed on problems requiring multiple geometry facts chained together (e.g., similar triangles inside a circle)
• Apply unit-circle values and the Pythagorean identity without a reference sheet
• Apply right-triangle trig (angle of elevation/depression, one trig ratio to find a missing side), reference angles and quadrant signs on the unit circle, and a printed Law of Sines or Law of Cosines on a non-right triangle
• Identify which geometry or trig archetypes you still miss most often

Hard ACT geometry and trig problems rarely require exotic knowledge. They require chaining facts together under time pressure. Trig is not a rare topic to dismiss: expect a small handful of trig questions per Math section, concentrated in the back half where the hard problems live. Most trig here is right triangles and the unit circle. When a non-right triangle appears, the question normally prints the Law of Sines or the Law of Cosines: use the Law of Cosines for two sides and the included angle (or for an angle when all three sides are known; the largest angle is opposite the longest side), the Law of Sines for two angles and a side, and match each side to its opposite angle before substituting. Typical shapes: (1) Inscribed shapes. A circle inside a square inside a triangle. Where you need to relate one shape's dimensions to another's. (2) Similar triangles hidden in a larger figure. The giveaway is parallel lines or shared angles. (3) Composite 3D solids that require breaking into basic pieces for volume or surface area. (4) Unit-circle questions asking for sin or cos of angles like 5π/6 or 4π/3. You must know the reference angle and quadrant signs. (5) Parabola and circle transformations where you must read the equation and describe the shift. (6) Trig identity simplifications using sin²θ + cos²θ = 1. (7) An inverse trig value inside a trig function, such as cos(sin⁻¹(8/17)): draw a right triangle with opposite side 8 and hypotenuse 17, find the third side, 15, and read cos = 15/17. (8) A circle measured through a chord: a 90° central angle makes an isosceles right triangle with two radii, so a chord of 6√2 means a radius of 6. Strategies: always draw the diagram, even if one is already given. Redraw it on scratch paper and label everything you can infer. On unit-circle questions, sketch the unit circle and mark the angle's quadrant before computing, so you get the sign right (sine is positive in quadrants 1 and 2, cosine in 1 and 4, tangent in 1 and 3). For similar-triangle problems, write the ratio of corresponding sides as a fraction and cross-multiply. Do not try to reason in your head. For composite volumes, write each piece's volume on a separate line before adding. As with any drill day: time it strictly, review the same day, and log your error types. Hard geometry/trig drills tend to expose formula gaps. If you miss three questions for not remembering the sector-area or cone-volume formulas, that is your drill target for the next session.

• Redraw every diagram on your scratch paper and label every length and angle you know. Most hard geometry questions require one label you did not notice at first.
• For unit-circle values, sketch the circle and identify the quadrant. Get the sign right first, then the value.
• For similar triangles, write the proportion as a fraction and cross-multiply. Do not do ratios in your head.
• If you blank on a formula, try picking numbers. Sometimes you can verify an answer choice by testing a simple case.
• Log missed formulas during review. Next session, start by writing the missed formulas on scratch paper.