ACT Math: Trigonometry - SOH CAH TOA, Right Triangle Ratios, Finding Sides and Angles

What This Is and Why It Matters for the ACT

Trigonometry: SOH CAH TOA is a fundamental concept in the ACT Math section. It appears in approximately 10-15% of the Math questions, with a moderate difficulty level. Understanding right triangle ratios will help you solve various problems involving triangles, circles, and other geometric shapes.

Key Concepts (What You Must Know)

• SOH CAH TOA: A mnemonic device to remember the trigonometric ratios:
  • Sine = Opposite over Hypotenuse
  • Cosine = Adjacent over Hypotenuse
  • Tangent = Opposite over Adjacent
• Right Triangle: A triangle with one 90-degree angle.
• Hypotenuse: The longest side of a right triangle, opposite the 90-degree angle.
• Opposite: The side of a triangle opposite a given angle.
• Adjacent: The side of a triangle adjacent to a given angle.

Step-by-Step Strategy for This Topic

1. Read the question carefully: Identify the type of triangle (right, isosceles, etc.) and the given information (angles, side lengths, etc.).
2. Determine the trigonometric ratio: Decide which ratio to use (sine, cosine, or tangent) based on the given information.
3. Eliminate wrong answers: Use the process of elimination to eliminate answer choices that are clearly incorrect.
4. Check your work: Verify your answer by plugging it back into the original equation or by using a calculator (if allowed).
5. Time management: Allocate 1-2 minutes per question, depending on the complexity.

Common mistake: Forgetting to check units: Make sure the units of the answer match the units of the question.

How It’s Tested on the ACT

• Math: Multiple-choice questions with five answer choices, often involving right triangles and trigonometric ratios.
• Distractors: Be careful of answer choices that are close but not quite correct, such as using the wrong trigonometric ratio or forgetting to square the hypotenuse.

Common Mistakes & Exam Traps

• The mistake: Forgetting to square the hypotenuse.
• Why it happens: Misunderstanding the Pythagorean theorem or rushing through the problem.
• How to avoid it: Double-check the units and make sure to square the hypotenuse when necessary.
• Exam board insight: The ACT examiners will penalize you for incorrect units or forgetting to square the hypotenuse.
• The mistake: Using the wrong trigonometric ratio.
• Why it happens: Misreading the question or misunderstanding the given information.
• How to avoid it: Read the question carefully and determine the correct trigonometric ratio based on the given information.
• Exam board insight: The ACT examiners will penalize you for using the wrong trigonometric ratio.
• The mistake: Forgetting to check units.
• Why it happens: Rushing through the problem or not double-checking the units.
• How to avoid it: Double-check the units and make sure they match the units of the question.
• Exam board insight: The ACT examiners will penalize you for incorrect units.

Practice Questions (3-5 questions)

Question 1: In a right triangle, the length of the hypotenuse is 10 inches and the length of the opposite side is 6 inches. What is the sine of the angle opposite the hypotenuse? Options: A) 0.6, B) 0.8, C) 0.9, D) 1.2, E) 1.5 Answer: B) 0.8 Explanation: Using the SOH CAH TOA mnemonic, we know that sine = opposite over hypotenuse. Plugging in the values, we get sine = 6/10 = 0.6. However, the question asks for the sine of the angle opposite the hypotenuse, which is actually 0.8.

Question 2: In a right triangle, the length of the adjacent side is 8 inches and the length of the hypotenuse is 10 inches. What is the cosine of the angle adjacent to the hypotenuse? Options: A) 0.4, B) 0.6, C) 0.8, D) 1.2, E) 1.5 Answer: C) 0.8 Explanation: Using the SOH CAH TOA mnemonic, we know that cosine = adjacent over hypotenuse. Plugging in the values, we get cosine = 8/10 = 0.8.

Question 3: In a right triangle, the length of the opposite side is 12 inches and the length of the adjacent side is 5 inches. What is the tangent of the angle opposite the adjacent side? Options: A) 1.2, B) 1.5, C) 2.4, D) 3.0, E) 4.0 Answer: C) 2.4 Explanation: Using the SOH CAH TOA mnemonic, we know that tangent = opposite over adjacent. Plugging in the values, we get tangent = 12/5 = 2.4.

Quick Reference Card (60-Second Summary)

• SOH CAH TOA: A mnemonic device to remember the trigonometric ratios.
• Right Triangle: A triangle with one 90-degree angle.
• Hypotenuse: The longest side of a right triangle, opposite the 90-degree angle.
• Opposite: The side of a triangle opposite a given angle.
• Adjacent: The side of a triangle adjacent to a given angle.
• Pythagorean Theorem: a^2 + b^2 = c^2, where c is the hypotenuse.

If You Get Stuck on Test Day

• Don't panic: Take a deep breath and read the question carefully again.
• Eliminate wrong answers: Use the process of elimination to eliminate answer choices that are clearly incorrect.
• Check your work: Verify your answer by plugging it back into the original equation or by using a calculator (if allowed).
• Skip and come back: If you're stuck, skip the question and come back to it later.

Related ACT Topics

• Pythagorean Theorem: a^2 + b^2 = c^2, where c is the hypotenuse.
• Circle Geometry: Understanding the properties of circles, including the circumference and area.
• Sine, Cosine, and Tangent: Understanding the relationships between these trigonometric ratios.