ACT Prep: Trigonometry (SOHCAHTOA, Radians, Graphs, Laws of Sines/Cosines)

ACT – Trigonometry (SOHCAHTOA, Radians, Graphs, Laws of Sines/Cosines)

ACT Trigonometry Study Guide

SOHCAHTOA, Radians, Graphs, Laws of Sines/Cosines

What This Is

Trigonometry on the ACT tests your ability to work with right and non-right triangles, unit circle concepts, and trigonometric graphs. You’ll need to apply SOHCAHTOA, convert between degrees and radians, interpret sine/cosine graphs, and use the Laws of Sines and Cosines to solve for missing sides or angles. A typical ACT question might ask: "In a right triangle, if the hypotenuse is 10 and the angle opposite a leg is 30°, what is the length of that leg?" (Answer: 5, using sin(30°) = ½). These skills are essential for geometry, physics, and real-world problems like calculating distances or angles in navigation, engineering, or architecture.

Key Terms & Rules

• SOHCAHTOA: A mnemonic for right-triangle trig ratios:
• Sine (sin ?) = Opposite / Hypotenuse
• Cosine (cos ?) = Adjacent / Hypotenuse
• Tangent (tan ?) = Opposite / Adjacent

• Example: In a right triangle with legs 3 and 4, and hypotenuse 5, sin-= 3/5 if-is opposite the side of length 3.

• Special Right Triangles:

• 30-60-90 Triangle: Sides are in ratio 1 : ?3 : 2 (short leg : long leg : hypotenuse).

• 45-45-90 Triangle: Sides are in ratio 1 : 1 : ?2 (legs : hypotenuse).

• Radians vs. Degrees:

• 180° =-radians (conversion: multiply degrees by ?/180 or radians by 180/?).

• Example: 60° = ?/3 radians.

• Unit Circle:

• A circle with radius 1 centered at the origin. Coordinates at angle-are (cos ?, sin ?).

• Key angles: 0°, 30° (?/6), 45° (?/4), 60° (?/3), 90° (?/2).

• Amplitude & Period of Trig Graphs:

• y = A sin(Bx + C) + D or y = A cos(Bx + C) + D:

  • Amplitude = |A| (height from midline to peak).
  • Period = 2?/|B| (horizontal length of one full cycle).
  • Phase Shift = -C/B (horizontal shift).
  • Vertical Shift = D (midline = y = D).

• Law of Sines:

• a/sin(A) = b/sin(B) = c/sin(C) (works for any triangle, not just right triangles).

• Use when: You have two angles and one side (AAS/ASA) or two sides and a non-included angle (SSA).

• Law of Cosines:

• c² = a² + b² - 2ab cos(C) (generalized Pythagorean theorem for non-right triangles).

• Use when: You have two sides and the included angle (SAS) or three sides (SSS).

• Pythagorean Identity:

• sin²? + cos²? = 1 (derived from the unit circle).

• Reciprocal Trig Functions:

• csc-= 1/sin ?, sec-= 1/cos ?, cot-= 1/tan ?.

• Inverse Trig Functions:

• sin?¹(x), cos?¹(x), tan?¹(x) return angles (e.g., sin?¹(½) = 30°).

Step-by-Step / Process Flow

How to Solve a Right-Triangle Trig Problem (SOHCAHTOA)

1. Label the triangle: Identify the hypotenuse, opposite side, and adjacent side relative to the given angle.
2. Choose the correct ratio: Use SOHCAHTOA to pick sin, cos, or tan based on the given sides.
3. Plug in values: Substitute the known side lengths into the ratio.
4. Solve for the unknown: Use algebra or a calculator (in degree mode for ACT).
5. Check units: Ensure your answer is in the correct units (e.g., degrees vs. radians).

Example: In a right triangle, if-= 40° and the adjacent side is 6, find the opposite side. - Step 1: Label: adjacent = 6, opposite = ?, hypotenuse = ?. - Step 2: Use tan(40°) = opposite/6. - Step 3: opposite = 6 tan(40°)-5.03.

How to Solve a Non-Right Triangle Problem (Laws of Sines/Cosines)

1. Draw the triangle: Sketch and label all known sides/angles.
2. Determine the law to use:
3. Law of Sines: If you have two angles and a side (AAS/ASA) or two sides and a non-included angle (SSA).
4. Law of Cosines: If you have two sides and the included angle (SAS) or three sides (SSS).
5. Write the equation: Plug known values into the law.
6. Solve for the unknown: Use algebra or a calculator.
7. Check for ambiguity (SSA): If using Law of Sines with SSA, there may be two possible solutions (ACT loves this trap!).

Example: In triangle ABC, A = 30°, B = 45°, and a = 8. Find side b. - Step 1: Draw triangle with angles 30°, 45°, and 105° (since 180° - 30° - 45° = 105°). - Step 2: Use Law of Sines (AAS). - Step 3: 8/sin(30°) = b/sin(45°)-b = 8 sin(45°)/sin(30°)-11.31.

How to Convert Between Degrees and Radians

1. Identify the conversion needed: Degrees-radians or radians-degrees?
2. Multiply by the conversion factor:
3. Degrees-radians: × (?/180).
4. Radians-degrees: × (180/?).
5. Simplify: Cancel-if possible (e.g., 180° =-radians).

Example: Convert 135° to radians. - 135° × (?/180) = 3?/4 radians.

Common Mistakes

• Mistake: Using the wrong trig ratio (e.g., using cos instead of sin).

• Correction: Always label the triangle and match the ratio to the sides you have. Why? SOHCAHTOA is side-specific!

• Mistake: Forgetting to set the calculator to degree mode (ACT uses degrees by default).

• Correction: Double-check mode before calculating. Why? Radians will give wrong answers for degree-based questions.

• Mistake: Ignoring the ambiguous case (SSA) in Law of Sines.

• Correction: If given two sides and a non-included angle, check if two triangles are possible. Why? The ACT often includes this as a distractor.

• Mistake: Misidentifying the amplitude or period in a trig graph.

• Correction: For y = A sin(Bx + C) + D, amplitude = |A|, period = 2?/|B|. Why? The ACT tests these transformations frequently.

• Mistake: Confusing Law of Sines and Law of Cosines.

• Correction: Use Law of Sines for angles/sides in a ratio; use Law of Cosines for sides/angles in a quadratic form. Why? The ACT tests when to apply each law.

Exam Insights

• Most-Tested Concepts:
• SOHCAHTOA (right-triangle trig) appears in ~50% of trig questions.
• Law of Sines/Cosines (non-right triangles) appears in ~30% of trig questions.

• Unit circle values (e.g., sin(30°), cos(?/4)) are tested in ~20% of questions.

• Tricky Distinctions:

• Degrees vs. radians: The ACT may give angles in radians but expect answers in degrees (or vice versa). Always check the question!
• Inverse trig functions: sin?¹(x) is not the same as 1/sin(x). The ACT tests this distinction.

• Graph transformations: The ACT often shifts or reflects sine/cosine graphs (e.g., y = -2 sin(x - ?/2)).

• Common Distractors:

• SSA ambiguity: The ACT may give two possible answers (e.g., "Which of the following could be the length of side b?").
• Pythagorean theorem misuse: Students forget that Law of Cosines is needed for non-right triangles.
• Sign errors: Forgetting that cos(120°) = -½ (negative in Quadrant II).

Quick Check Questions

1. In a right triangle, if-= 25° and the hypotenuse is 12, what is the length of the side opposite
2. A) 5.07
3. B) 10.88
4. C) 11.28
5. D) 13.24

6. Answer: A) 5.07 (sin(25°) = opposite/12-opposite = 12 sin(25°)-5.07).

7. In triangle ABC, a = 7, b = 10, and C = 50°. What is the length of side c?

8. A) 7.8
9. B) 8.9
10. C) 9.5
11. D) 10.2

12. Answer: A) 7.8 (Law of Cosines: c² = 7² + 10² - 2(7)(10)cos(50°)-60.8-c-7.8).

13. What is the amplitude and period of y = 3 cos(2x - ?) + 1?

14. Answer: Amplitude = 3, Period =? (Amplitude = |3|, Period = 2?/2 = ?).

Last-Minute Cram Sheet

1. SOHCAHTOA: sin = opp/hyp, cos = adj/hyp, tan = opp/adj.
2. 30-60-90: 1 : ?3 : 2; 45-45-90: 1 : 1 : ?2.
3. 180° =-radians-convert with ?/180 or 180/?.
4. Law of Sines: a/sin(A) = b/sin(B) = c/sin(C) (use for AAS/ASA/SSA).
5. Law of Cosines: c² = a² + b² - 2ab cos(C) (use for SAS/SSS).
6. Unit circle: (cos ?, sin ?); sin(30°) = ½, cos(60°) = ½.
7. Amplitude = |A|, Period = 2?/|B| for y = A sin(Bx + C) + D.
8. SSA ambiguity: Law of Sines may give two solutions (check if the angle is acute/obtuse).
9. Calculator mode: Always use degree mode unless the question specifies radians.
10. Inverse trig: sin?¹(x)-1/sin(x) (it’s the angle whose sine is x).