Trigonometry

• Use SOH-CAH-TOA to find missing sides and angles in right triangles, including angle of elevation and depression problems
• Convert between degrees and radians, and rebuild unit-circle values from the 30-60-90 and 45-45-90 triangles plus the quadrant signs
• Apply sin²θ + cos²θ = 1 and tan θ = sin θ / cos θ, and substitute into the Law of Sines or Law of Cosines when a question prints it
• Match a sine or cosine graph to its equation by checking the amplitude, period, and midline against each choice
• Finish on one ACT-hard question and say what made it hard

SOH-CAH-TOA defines the three main trig ratios: sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent. These ratios let you find unknown sides or angles in right triangles. Radians are another way to measure angles: 180° = π radians, so to convert, multiply degrees by π/180. The identity sin²θ + cos²θ = 1 is the most important trig identity on the ACT. Non-right triangles: when the ACT asks about a triangle with no right angle, the question normally prints the Law of Sines (a/sin A = b/sin B = c/sin C) or the Law of Cosines (c² = a² + b² − 2ab·cos C), where each lowercase side is opposite the capital angle with the same letter. The job is not to memorize them but to recognize when they apply and substitute correctly. Recognition cue: no right angle, and you know two sides and the angle between them (use the Law of Cosines to find the third side) or two angles and a side (use the Law of Sines). Label every side with the letter of the angle across from it before substituting. Worked example: two sides are 5 and 8 and the angle between them is 60°, so the third side c satisfies c² = 5² + 8² − 2(5)(8)cos 60° = 25 + 64 − 40 = 49, and c = 7. The applications to practice are angle-of-elevation and angle-of-depression word problems (draw the right triangle, then use SOH-CAH-TOA to find the missing side or angle), and converting between degrees and radians. Unit circle without memorizing a table: the point at angle θ on a circle of radius 1 centered at the origin is (cos θ, sin θ), and the benchmark values come from the two special right triangles with hypotenuse 1: the 30-60-90 triangle gives 1/2 and √3/2, and the 45-45-90 triangle gives √2/2 for both legs. For any other angle, find the reference angle (the acute angle to the x-axis), take the value from the triangle, then attach the sign for the quadrant: sine is positive in Quadrants I and II, cosine in I and IV, and tangent in I and III. For example, 150° (5π/6 radians) has reference angle 30°, so sin 150° = 1/2 and cos 150° = −√3/2. To match a graph to y = A sin(Bx) + D or y = A cos(Bx) + D, check one feature at a time against every choice: the midline is y = D, the amplitude is |A| (the distance from the midline to a maximum), and the period is 2π/|B|; y = A sin(Bx) + D is on its midline at x = 0, while y = A cos(Bx) + D is at a maximum there when A > 0. Trigonometry accounts for a small handful of questions, and they tend to appear in the second half of the Math section among the harder questions. The hardest questions, at the end of the section: an identity hidden inside a larger expression, such as (1 − cos²θ)/sin θ, which equals sin θ wherever sin θ is not 0 (check with a friendly angle such as 30°), and unit-circle values for angles in radians, negative, or past a full turn, where the reference angle and the quadrant decide the value and its sign. Also late in the section: an inverse-trig composition such as cos(Arcsin(3/5)), where Arcsin(3/5) is an acute angle with opposite side 3 and hypotenuse 5, so the adjacent side is 4 and the value is 4/5; and a printed Law of Cosines where the job is to pick the right equation, remembering that the smallest angle is opposite the shortest side and the largest angle opposite the longest.

• Memorize SOH-CAH-TOA and write it at the top of your scratch paper at the start of the test.
• If a triangle has no right angle and the question prints the Law of Sines or Law of Cosines, label each side with the letter of its opposite angle, then substitute. Otherwise, look for an altitude that creates right triangles.
• Know the trig values for common angles: sin 30° = 1/2, cos 30° = √3/2, sin 45° = cos 45° = √2/2.