Trigonometry

What you'll learn

• Define sine, cosine, and tangent using right triangle sides
• Solve for missing sides and angles in right triangles using trig ratios
• Understand radian measure and convert between degrees and radians
• Apply basic trig identities: sin²θ + cos²θ = 1, tanθ = sinθ/cosθ
• Solve right-triangle problems the ACT actually tests: angle of elevation and depression, finding an unknown side or angle from one trig ratio, and combining SOH-CAH-TOA with the Pythagorean theorem

Key concepts

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. Note: the ACT does NOT test the Law of Sines or the Law of Cosines, so do not spend time on them. What the ACT does test is right-triangle applications: 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 occasional unit-circle basics (converting degrees and radians, and knowing the sign of sine and cosine in each quadrant, for example sin(5π/6) is positive). Trigonometry usually accounts for a small handful of questions (often around four), and they tend to appear in the second half of the test among the harder questions.

Pro tips

• Memorize SOH-CAH-TOA and write it at the top of your scratch paper at the start of the test.
• If a trig question involves a non-right triangle, look for a way to draw an altitude to create right triangles.
• Know the trig values for common angles: sin 30° = 1/2, cos 30° = √3/2, sin 45° = cos 45° = √2/2.